📖Topic Explanations

🌐 Overview
Hello students! Welcome to the fascinating world of Ordinary Differential Equations (ODEs)! Prepare to unlock the mathematical secrets behind change, motion, and growth that shape our universe.



Have you ever paused to think about how scientists predict the spread of a disease, engineers design the flight path of a rocket, or economists model market fluctuations? The answer lies often within the realm of Ordinary Differential Equations. These aren't just abstract mathematical constructs; they are the language we use to describe systems where quantities are constantly changing.



At its core, a differential equation is an equation that involves an unknown function and its derivatives. When these derivatives are with respect to a single independent variable, we call them Ordinary Differential Equations. They are powerful tools that allow us to translate real-world dynamic processes into mathematical models, which can then be solved to understand past behaviour and predict future outcomes.



Before we dive deep into solving these intriguing equations, we first need to understand their fundamental characteristics. Just like classifying different types of vehicles before learning to drive them, we need to classify ODEs. This is where the concepts of order and degree come into play.




  • The order of an Ordinary Differential Equation tells us the highest derivative present in the equation. It's a fundamental property that dictates the complexity and the general approach we'll take to find its solution.

  • The degree, on the other hand, relates to the power of the highest order derivative, once the equation has been made free of fractions and radicals with respect to the derivatives. This characteristic further refines our understanding of the ODE's structure and can influence solution techniques.



Think of order and degree as the 'DNA' of a differential equation. They are crucial identifiers that not only help us categorize ODEs but also guide us towards the appropriate methods for finding their solutions. Understanding these concepts is the first, indispensable step towards mastering ODEs.



For your JEE Main and board examinations, identifying the order and degree of a given differential equation is a frequently tested skill. It's a foundational concept that forms the bedrock for more advanced topics in differential equations. In this section, you will learn to precisely define and confidently identify these crucial characteristics for a wide range of Ordinary Differential Equations.



So, get ready to lay a strong foundation in this vital area of mathematics. Let’s embark on this exciting journey to unravel the dynamics of the world around us!

📚 Fundamentals
Hey everyone! Welcome to the fascinating world of Differential Equations! Think of mathematics as a language that describes the universe around us. We've used numbers, variables, and functions to describe static things or simple relationships. But what about things that *change*? The speed of a car, the growth of a population, the flow of heat, the way a disease spreads – all these involve change over time or space. And that's where differential equations come in!

### 1. The Language of Change: What is a Differential Equation?

Before we dive into the nitty-gritty, let's establish a common ground. You're familiar with equations like:
* `x + 5 = 10` (an algebraic equation)
* `sin(x) = 0.5` (a trigonometric equation)
* `y = x^2 + 2x` (a functional relationship)

These equations help us find unknown values or describe relationships between variables.

Now, let's talk about change. How do we represent change mathematically? We use derivatives! Remember `dy/dx` or `f'(x)`? It tells us the instantaneous rate of change of a function `y` with respect to `x`. If `y` is the position of a car and `x` is time, then `dy/dx` is its velocity. If `dy/dx` is velocity, then `d^2y/dx^2` (or `y''`) is the acceleration!

So, what happens when an equation doesn't just involve `x` and `y`, but also their derivatives? Bingo! You get a Differential Equation.


Definition: A differential equation is an equation that involves an unknown function and one or more of its derivatives.



These equations are incredibly powerful because they allow us to model dynamic situations. For instance:
* Newton's second law of motion (`F = ma`) can be written as a differential equation if `a` is `d^2x/dt^2`.
* The rate at which a population grows often depends on the current population size, leading to a differential equation.
* The cooling of a hot object in a room follows a differential equation (Newton's Law of Cooling).

It's like saying, "The way this thing is changing is related to its current state in this particular mathematical way."

### 2. Ordinary vs. Partial Differential Equations (ODEs vs. PDEs)

Okay, so we know what a differential equation is. Now, let's classify them into two big families: Ordinary and Partial. For our JEE journey, we'll primarily focus on Ordinary Differential Equations (ODEs).

#### 2.1 Ordinary Differential Equations (ODEs)

Imagine a scenario where the quantity you're interested in (say, `y`) depends only on one single independent variable (like `x` or time `t`).
For example, if the temperature of a coffee cup (`T`) depends only on time (`t`), then we'd use derivatives like `dT/dt`.


Definition: An Ordinary Differential Equation (ODE) is a differential equation involving derivatives of one or more dependent variables with respect to a single independent variable.



The key here is "single independent variable". This means all the derivatives in an ODE are ordinary derivatives (like `dy/dx`, `d^2y/dx^2`, etc.), not partial derivatives.

Examples of ODEs:
1. `dy/dx = x^2 + y`
2. `d^2y/dx^2 + 4(dy/dx) + 3y = sin(x)`
3. `m(d^2x/dt^2) = -kx` (Simple Harmonic Motion)
4. `y' + xy^2 = 0` (Here `y'` means `dy/dx`)

#### 2.2 Partial Differential Equations (PDEs)

Just for context, if a quantity depends on two or more independent variables, then we'd use partial derivatives. For example, the temperature (`T`) inside a metal plate might depend on both its position (`x`, `y`) and time (`t`). In such cases, you'd encounter terms like `∂T/∂x`, `∂T/∂y`, `∂T/∂t`. Equations involving these are called Partial Differential Equations (PDEs). We won't be dealing with PDEs in JEE Mains or Advanced, but it's good to know the distinction!

### 3. Understanding the "Order" of a Differential Equation

Okay, now let's get into the defining characteristics of an ODE. The first one is its "order". Think of "order" as telling you how many times you've differentiated the unknown function at most.


Definition: The order of a differential equation is the order of the highest derivative present in the equation.



It's pretty straightforward! You just look at all the derivatives in the equation and find the one with the highest order (e.g., first derivative `dy/dx`, second derivative `d^2y/dx^2`, third derivative `d^3y/dx^3`, and so on). The highest among them determines the equation's order.

Let's look at some examples:










































Differential Equation Derivatives Present Highest Order Derivative Order
`dy/dx = 3x + 2` `dy/dx` (1st order) `dy/dx` 1
`d^2y/dx^2 + 5(dy/dx) + 6y = 0` `d^2y/dx^2` (2nd order), `dy/dx` (1st order) `d^2y/dx^2` 2
`(dy/dx)^3 + y = x^2` `dy/dx` (1st order) `dy/dx` 1
`y''' + (y'')^2 + y' = cos(x)` `y'''` (3rd order), `y''` (2nd order), `y'` (1st order) `y'''` 3
`(d^4y/dx^4) + e^x (d^2y/dx^2) + 7y = 0` `d^4y/dx^4` (4th order), `d^2y/dx^2` (2nd order) `d^4y/dx^4` 4


Why is order important? The order of a differential equation often tells us about the number of arbitrary constants that will appear in its general solution. A first-order ODE will have one arbitrary constant, a second-order ODE will have two, and so on. This is crucial for solving them!

### 4. Understanding the "Degree" of a Differential Equation

Now for the second important characteristic: the "degree". This one can be a little trickier, so pay close attention!


Definition: The degree of a differential equation is the highest power (exponent) of the highest order derivative, AFTER the equation has been made free from radicals and fractions involving derivatives.



Let's break that down:

1. Find the highest order derivative: This is the same step as finding the order.
2. Clear radicals and fractions involving derivatives: This is the crucial step. If you see square roots (`sqrt`), cube roots, or derivatives in the denominator, you *must* eliminate them by raising both sides to a power or multiplying by appropriate terms. The goal is to make the equation look like a polynomial in terms of its derivatives.
3. Find the exponent: Once the equation is "cleaned up" (polynomial form with respect to derivatives), look at the highest power of that highest order derivative. That's your degree!

Important Note: If the differential equation cannot be expressed as a polynomial in its derivatives (e.g., if a derivative is inside a transcendental function like `sin(y')`, `e^(y'')`, `log(y''')`), then its degree is not defined.

Let's walk through some examples:



  1. Simple Case: No radicals or fractions

    Equation: `d^2y/dx^2 + 5(dy/dx) + 6y = 0`


    • Highest order derivative: `d^2y/dx^2` (order 2)

    • No radicals or fractions involving derivatives.

    • Power of `d^2y/dx^2`: `(d^2y/dx^2)^1`.

    • Degree: 1.






  2. Highest order derivative with a power

    Equation: `(d^3y/dx^3) + (dy/dx)^5 + y = x`


    • Highest order derivative: `d^3y/dx^3` (order 3)

    • No radicals or fractions.

    • Power of `d^3y/dx^3`: `(d^3y/dx^3)^1`.

    • Degree: 1.

    • Notice: The power `5` on `(dy/dx)` doesn't matter for the degree, because `dy/dx` is not the highest order derivative.






  3. Equation with a radical involving a derivative

    Equation: `sqrt(1 + (dy/dx)^2) = d^2y/dx^2`


    • First, clear the radical. Square both sides:

    • `1 + (dy/dx)^2 = (d^2y/dx^2)^2`

    • Highest order derivative: `d^2y/dx^2` (order 2)

    • Power of `d^2y/dx^2`: `(d^2y/dx^2)^2`.

    • Degree: 2.






  4. Equation with fractional powers of derivatives

    Equation: `(d^2y/dx^2)^(3/2) = (dy/dx)^4`


    • Clear the fractional power. Raise both sides to the power of 2:

    • `((d^2y/dx^2)^(3/2))^2 = ((dy/dx)^4)^2`

    • `(d^2y/dx^2)^3 = (dy/dx)^8`

    • Highest order derivative: `d^2y/dx^2` (order 2)

    • Power of `d^2y/dx^2`: `(d^2y/dx^2)^3`.

    • Degree: 3.






  5. Degree not defined case

    Equation: `sin(dy/dx) + y = x`


    • Highest order derivative: `dy/dx` (order 1)

    • Can this equation be expressed as a polynomial in `dy/dx`? No, because `dy/dx` is an argument of the `sin` function. You cannot remove the `sin` function to isolate `dy/dx` as a simple polynomial term.

    • Degree: Not defined.






  6. Another 'degree not defined' case

    Equation: `e^(d^2y/dx^2) + x(dy/dx) = 0`


    • Highest order derivative: `d^2y/dx^2` (order 2)

    • Can this be expressed as a polynomial in `d^2y/dx^2`? No, because `d^2y/dx^2` is in the exponent of `e`.

    • Degree: Not defined.







Why is degree important? The degree of a differential equation, especially if it's 1, can simplify the methods needed to solve it. First-degree differential equations are often linear and have well-established solution techniques. Higher degree equations can be more complex to solve.

### 5. CBSE vs. JEE Focus Callouts

* CBSE: For CBSE, understanding the definitions of order and degree, along with straightforward examples, is usually sufficient. The "degree not defined" cases are also important. The focus is on correctly identifying them from given equations.
* JEE Mains & Advanced: While the core definitions remain the same, JEE might present questions that require a bit more algebraic manipulation to determine the degree (like clearing radicals or fractional powers, or identifying 'degree not defined' cases quickly). Sometimes, the question might combine this with other concepts. Practice with a variety of forms is key!

### Summary

Let's quickly recap:
* A Differential Equation is an equation involving derivatives.
* An Ordinary Differential Equation (ODE) involves derivatives with respect to a single independent variable.
* The Order of an ODE is the order of its highest derivative.
* The Degree of an ODE is the power of its highest order derivative, after clearing any radicals or fractions involving derivatives. If it can't be made into a polynomial in its derivatives, the degree is undefined.

You've just taken a crucial first step in understanding differential equations. Knowing the order and degree is like identifying the "type" and "complexity level" of the equation, which then guides us on how to approach its solution. Keep practicing these definitions with various examples, and you'll master this fundamental concept in no time!
🔬 Deep Dive

Hello, future engineers and mathematicians! Welcome to this deep dive into the fascinating world of Differential Equations. Today, we're going to build a solid foundation by understanding two fundamental characteristics of Ordinary Differential Equations (ODEs): their Order and their Degree. These concepts are absolutely crucial, not just for your JEE preparation, but for comprehending how these equations are classified and solved.



Think of differential equations as the language of change. They describe how quantities change with respect to one another. From physics to economics, biology to engineering, differential equations are the backbone of mathematical modeling.



1. What are Differential Equations? (A Quick Recap)



At its core, a Differential Equation (DE) is an equation that involves an unknown function and one or more of its derivatives with respect to one or more independent variables. The goal is often to find this unknown function.



For instance, consider the simple equation for the velocity of an object falling under gravity: $dv/dt = g$. Here, $v$ is the velocity (our unknown function), $t$ is time (the independent variable), and $g$ is the acceleration due to gravity (a constant). This is a differential equation.



1.1 Ordinary vs. Partial Differential Equations



Before we dive into order and degree, it's essential to distinguish between the two main types of differential equations:




  • Ordinary Differential Equations (ODEs): These involve derivatives with respect to only one independent variable. For example, $dy/dx + y = x^2$. Here, $y$ is a function of a single independent variable, $x$. In JEE Mains & Advanced, our primary focus is on ODEs.




  • Partial Differential Equations (PDEs): These involve partial derivatives with respect to two or more independent variables. For example, $partial u / partial x + partial u / partial y = 0$. Here, $u$ is a function of two independent variables, $x$ and $y$. PDEs are typically covered in higher mathematics courses and are beyond the scope of JEE.




For the remainder of this discussion, we will concentrate exclusively on Ordinary Differential Equations (ODEs).



2. Understanding the Order of a Differential Equation



The order of a differential equation is perhaps the more straightforward of the two concepts. It tells us about the highest derivative present in the equation. Let's formalize this.



2.1 Definition of Order



The Order of a differential equation is defined as the order of the highest derivative appearing in the equation.



In simpler terms, you scan the entire equation, identify all the derivatives present ($dy/dx$, $d^2y/dx^2$, $d^3y/dx^3$, etc.), and then pick the one with the largest "power" of differentiation. That "power" is the order.



Intuition and Analogy:


Imagine you're describing how something changes. If you just say "it's moving" ($dy/dx$), that's a first-order change (velocity). If you say "its movement is speeding up or slowing down" ($d^2y/dx^2$), that's a second-order change (acceleration). If you talk about how the acceleration itself is changing ($d^3y/dx^3$, known as jerk), that's a third-order change. The order of the differential equation tells you the "highest level of change" being described.



2.2 Examples of Determining Order





  1. Example 1: $dy/dx + y = cos x$


    Explanation: The only derivative present is $dy/dx$, which is a first-order derivative.
    Therefore, the order of this differential equation is 1.




  2. Example 2: $d^2y/dx^2 + 5(dy/dx)^3 - 2y = e^x$


    Explanation: We have two derivatives here: $dy/dx$ (first order) and $d^2y/dx^2$ (second order). The highest order derivative is $d^2y/dx^2$.
    Therefore, the order of this differential equation is 2.




  3. Example 3: $(d^3y/dx^3)^2 + x(dy/dx) = sin(d^4y/dx^4)$


    Explanation: The derivatives are $dy/dx$ (1st order), $d^3y/dx^3$ (3rd order), and $d^4y/dx^4$ (4th order). The highest among these is $d^4y/dx^4$. The fact that it's inside a sine function does not affect the order.
    Therefore, the order of this differential equation is 4.




  4. Example 4: $sqrt{1 + (dy/dx)^2} = frac{d^2y}{dx^2}$


    Explanation: Even though there's a square root, we first identify the derivatives: $dy/dx$ (1st order) and $d^2y/dx^2$ (2nd order). The highest is $d^2y/dx^2$. The presence of the radical does not alter the order.
    Therefore, the order of this differential equation is 2.




JEE Focus: Order is generally straightforward. The traps usually lie in confusing the *power* of a derivative with its *order*. Remember, $ (dy/dx)^3 $ is still a first-order derivative, just raised to the power of 3.



3. Understanding the Degree of a Differential Equation



The degree of a differential equation is a bit more nuanced and often a source of confusion. It requires careful attention to the algebraic form of the equation.



3.1 Definition of Degree



The Degree of a differential equation is defined as the highest power (exponent) of the highest order derivative, after the equation has been made free from radicals and fractions involving derivatives, and after it has been expressed as a polynomial in derivatives.



Key Conditions for Degree:




  1. Polynomial in Derivatives: The equation must be expressible as a polynomial in its derivatives. This means terms like $e^{dy/dx}$, $sin(d^2y/dx^2)$, $log(dy/dx)$ are NOT allowed. If such terms are present, the degree is undefined.




  2. Free from Radicals and Fractions of Derivatives: All derivatives must have integer powers. If you have terms like $sqrt{dy/dx}$ or $1/(d^2y/dx^2)$, you must clear them by raising both sides to a suitable power or multiplying by appropriate terms, respectively.




  3. Highest Power of Highest Order: Once the above conditions are met, locate the highest order derivative identified earlier. Its exponent (power) is the degree.





Intuition and Analogy:


If order tells you the "highest level of change", degree tells you "how complex that highest level of change is algebraically". Is the acceleration term just $d^2y/dx^2$? Or is it $(d^2y/dx^2)^3$? The exponent indicates the non-linearity or "polynomial complexity" of that highest derivative term.



3.2 Steps to Determine Degree





  1. Identify the Highest Order Derivative: As we did for finding the order, first pinpoint the highest derivative in the equation.




  2. Check for Transcendental Functions of Derivatives: Look for derivatives appearing as arguments of transcendental functions (e.g., $sin(dy/dx)$, $e^{d^2y/dx^2}$, $log(dy/dx)$). If any such term exists, the degree is undefined. Stop here.




  3. Clear Radicals/Fractions Involving Derivatives: If there are fractional powers (like $sqrt{...}$ which means power $1/2$) or derivatives in the denominator, manipulate the equation algebraically to make all powers of derivatives integers. This often involves raising both sides of the equation to a common power or multiplying by suitable terms.




  4. Find the Power of the Highest Order Derivative: Once the equation is a polynomial in its derivatives (i.e., conditions 2 & 3 are met), find the power of the highest order derivative term you identified in step 1. This power is the degree.





3.3 Examples of Determining Degree





  1. Example 1: $dy/dx + y = cos x$



    • Highest order derivative: $dy/dx$ (Order 1).

    • No radicals/fractions, no transcendental functions of derivatives.

    • Power of $dy/dx$ is 1.


    Therefore, the degree is 1.




  2. Example 2: $d^2y/dx^2 + 5(dy/dx)^3 - 2y = e^x$



    • Highest order derivative: $d^2y/dx^2$ (Order 2).

    • No radicals/fractions, no transcendental functions of derivatives.

    • The power of $d^2y/dx^2$ is 1. (Note: The power of $dy/dx$ is 3, but $dy/dx$ is not the highest order derivative).


    Therefore, the degree is 1.




  3. Example 3: $(d^3y/dx^3)^2 + x(dy/dx) = 0$



    • Highest order derivative: $d^3y/dx^3$ (Order 3).

    • No radicals/fractions, no transcendental functions of derivatives.

    • The power of $d^3y/dx^3$ is 2.


    Therefore, the degree is 2.




  4. Example 4: $sqrt{1 + (dy/dx)^2} = frac{d^2y}{dx^2}$



    • Highest order derivative: $d^2y/dx^2$ (Order 2).

    • There's a radical involving a derivative. We need to clear it.
      Square both sides: $1 + (dy/dx)^2 = (d^2y/dx^2)^2$

    • Now the equation is free from radicals. The highest order derivative is $d^2y/dx^2$. Its power is 2.


    Therefore, the degree is 2.




  5. Example 5: $(d^2y/dx^2)^{3/2} = (dy/dx) + x$



    • Highest order derivative: $d^2y/dx^2$ (Order 2).

    • There's a fractional power ($3/2$) involving a derivative. Clear it by squaring both sides:
      $((d^2y/dx^2)^{3/2})^2 = ((dy/dx) + x)^2$
      $(d^2y/dx^2)^3 = ((dy/dx) + x)^2$

    • Now the equation is free from fractional powers. The highest order derivative is $d^2y/dx^2$. Its power is 3.


    Therefore, the degree is 3.




  6. Example 6: $x(d^3y/dx^3) + y(dy/dx)^4 = log(d^2y/dx^2)$



    • Highest order derivative: $d^3y/dx^3$ (Order 3).

    • However, notice the term $log(d^2y/dx^2)$. The derivative $d^2y/dx^2$ is inside a logarithmic function. This means the equation cannot be expressed as a polynomial in its derivatives.


    Therefore, the degree is undefined.




  7. Example 7: $e^{dy/dx} + x^2 = 0$



    • Highest order derivative: $dy/dx$ (Order 1).

    • The derivative $dy/dx$ is in the exponent of $e$. This is a transcendental function of a derivative.


    Therefore, the degree is undefined.





4. Summary: Order vs. Degree



Let's consolidate our understanding with a comparison table:
































Feature Order Degree
Definition Highest derivative present in the equation. Highest power of the highest order derivative, after clearing radicals and fractions, and ensuring it's a polynomial in derivatives.
Always Defined? Yes (for any differential equation). No, can be undefined if not a polynomial in derivatives (e.g., $e^{y'}$, $sin(y'')$).
Requires Algebraic Manipulation? Rarely, usually by inspection. Often, to clear fractional/negative powers of derivatives.
Focus Level of differentiation. Power of the most significant derivative term.


5. Advanced Considerations and JEE Traps



JEE Advanced Focus: While CBSE problems on order and degree are usually direct, JEE often tests your understanding of the conditions for degree. Here are some critical points:




  • Implicit Form: Differential equations can be given in implicit forms. Always try to isolate and clarify the derivative terms, especially when dealing with radicals or fractions.




  • Transcendental Functions: This is the biggest trap! If any derivative, of any order, is an argument to a non-algebraic function (like sine, cosine, log, exponential, inverse trig functions), the degree is undefined. Examples: $sin(dy/dx)$, $e^{d^2y/dx^2}$, $log(y')$. Make sure you remember this rule.


    For example, in $sin(dy/dx) = x+y$, the order is 1 (due to $dy/dx$), but the degree is undefined because $dy/dx$ is inside a sine function.


    However, if the function is just multiplied by a derivative, it's fine. For example, $(dy/dx) sin y = x^2$ has order 1, degree 1.




  • Simplification is Key: Always simplify the equation to its polynomial form with integer powers before determining the degree. Don't jump to conclusions.





Example of a Tricky JEE Problem:


Determine the order and degree of the differential equation: $(1 + y_1^2)^{3/2} = y_2$ where $y_1 = dy/dx$ and $y_2 = d^2y/dx^2$.


Step-by-step Solution:




  1. Identify all derivatives: We have $y_1 = dy/dx$ (1st order) and $y_2 = d^2y/dx^2$ (2nd order).




  2. Determine Order: The highest order derivative is $y_2 = d^2y/dx^2$.
    So, the Order is 2.




  3. Check for Undefined Degree Conditions: No derivatives are inside transcendental functions.




  4. Clear Radicals/Fractions for Degree:
    The equation is $(1 + (dy/dx)^2)^{3/2} = d^2y/dx^2$.
    To clear the fractional power $3/2$, we need to square both sides:
    $((1 + (dy/dx)^2)^{3/2})^2 = (d^2y/dx^2)^2$
    $(1 + (dy/dx)^2)^3 = (d^2y/dx^2)^2$




  5. Find the power of the highest order derivative:
    After algebraic manipulation, the highest order derivative is $d^2y/dx^2$. Its power is 2.
    So, the Degree is 2.





This kind of problem is typical for JEE where a little algebraic manipulation is required before identifying the degree.



Understanding order and degree is not just about memorizing definitions; it's about correctly interpreting the structure of differential equations. This fundamental knowledge will be vital as you progress to solving these equations in various methods like variable separable, homogeneous, linear differential equations, and exact differential equations.

🎯 Shortcuts

Grasping the concepts of order and degree of a differential equation is fundamental. These definitions are direct applications, but often lead to confusion due to specific conditions, especially for degree. Using mnemonics and short-cuts can help you recall these rules quickly and accurately under exam pressure.



Order of a Differential Equation: Mnemonic & Short-cut


The order of a differential equation is the order of the highest derivative present in the equation.



  • Mnemonic: "ORDer is the Highest ORDinal (position) of the derivative you find."

  • Explanation: Think of "ordinal" as "first, second, third..." This directly tells you to scan the equation for all derivatives and pick the largest "ordinal number" associated with any derivative. For example, if you see d³y/dx³, its order is 3. If the highest is d⁵y/dx⁵, the order is 5.

  • JEE/CBSE Tip: Determining the order is generally straightforward. It's the simplest part of this topic and rarely causes issues.



Degree of a Differential Equation: Mnemonic & Short-cut


The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in its derivatives. This means it must be free from radicals and fractional powers of derivatives.



  • Mnemonic: For Degree, remember 'R.F.P.H.':

    • Remove Fractional/Radical powers (by appropriate algebraic steps like squaring, cubing, etc.).

    • Then, find the Power of the Highest order derivative.



  • Explanation: This mnemonic highlights the critical pre-condition. Many students directly look for the power without ensuring the equation is a polynomial in derivatives. The 'R.F.P.H.' sequence ensures you perform the necessary algebraic manipulation first. If, even after algebraic manipulation, the equation cannot be expressed as a polynomial in its derivatives (e.g., terms like sin(dy/dx), e^(d²y/dx²), log(d³y/dx³)), then the degree is not defined.

  • JEE/CBSE Warning: This is where most mistakes occur. Always ensure the equation is a polynomial in its derivatives before stating the degree. If it's not, the degree is UNDEFINED.



Quick Check Summary Table

























Concept Mnemonic/Short-cut Key Action Critical Note
Order Highest ORDinal Identify the derivative with the highest differentiation count. Always defined.
Degree R.F.P.H.

  1. Remove Fractional/Radical powers.

  2. Find Power of the Highest order derivative.


 Defined only if the equation is a polynomial in its derivatives. Else, it's Undefined.


Practical Example for Mnemonics


Consider the differential equation: (1 + (dy/dx)²)³/² = 5 (d²y/dx²)




  1. Order:

    • Derivatives present: dy/dx (order 1) and d²y/dx² (order 2).

    • Highest Ordinal derivative: d²y/dx².

    • Therefore, Order = 2.




  2. Degree:

    • Apply 'R.F.P.H.':

    • Remove Fractional/Radical powers: The ³/² power needs to be removed. Square both sides:
      (1 + (dy/dx)²)³ = (5 (d²y/dx²))²
      (1 + (dy/dx)²)³ = 25 (d²y/dx²)²

    • Now the equation is a polynomial in its derivatives.

    • Find the Power of the Highest order derivative: The highest order derivative is d²y/dx², and its power in the modified equation is 2.

    • Therefore, Degree = 2.




Mastering these simple mnemonics will save you time and prevent common errors in exams. Stay sharp and practice regularly!

💡 Quick Tips

Mastering the identification of order and degree is fundamental for solving differential equations and is a common scoring point in both JEE Main and board exams. Here are quick tips to ensure you get it right every time:



Quick Tips for Order



  • The order of a differential equation is the order of the highest derivative appearing in the equation.

  • It is always a positive integer.

  • Identifying the order is generally straightforward. Simply scan the equation for terms like dy/dx, d²y/dx², d³y/dx³, etc., and pick the highest one.

  • Example: For (d³y/dx³) + x(dy/dx)² + y = 0, the highest derivative is d³y/dx³, so the order is 3.



Quick Tips for Degree



  • The degree of a differential equation is the power of the highest order derivative, *after* the equation has been made free from radicals and fractional powers of the derivatives.

  • The degree is defined only if the differential equation can be expressed as a polynomial in its derivatives.

  • It is always a positive integer.

  • CRITICAL CONDITION: Before determining the degree, ensure that the differential equation is a polynomial in its derivatives. This means:

    • No fractional powers of derivatives (e.g., √(dy/dx), (d²y/dx²)³ᐟ²).

    • No derivatives within trigonometric functions (e.g., sin(dy/dx)).

    • No derivatives within exponential functions (e.g., e^(dy/dx)).

    • No derivatives within logarithmic functions (e.g., log(d²y/dx²)).



  • If any of these critical conditions are not met, the degree is undefined.

  • Strategy for Fractional Powers: If there are fractional powers involving derivatives, square/cube/raise to an appropriate power on both sides of the equation to eliminate them. Only *then* determine the degree.

  • Example 1 (Polynomial in derivatives): For (d³y/dx³) + x(dy/dx)² + y = 0, the highest order derivative is d³y/dx³, and its power is 1. So, the degree is 1.

  • Example 2 (Fractional power): For (d²y/dx²) = √(1 + (dy/dx)²):

    • First, eliminate the fractional power: Square both sides.

    • (d²y/dx²)² = 1 + (dy/dx)²

    • Now, the highest order derivative is d²y/dx², and its power is 2. So, the degree is 2.



  • Example 3 (Not a polynomial): For sin(dy/dx) + x = 0:

    • The derivative dy/dx is inside a trigonometric function.

    • The equation cannot be expressed as a polynomial in derivatives.

    • Therefore, the degree is undefined. The order is 1.





JEE Main vs. CBSE Boards



  • Both JEE Main and CBSE boards test these concepts.

  • CBSE: Typically involves simpler differential equations where the polynomial form is obvious or requires a single step (like squaring) to resolve fractional powers.

  • JEE Main: May present equations that are initially disguised, requiring careful manipulation to determine if the polynomial condition is met, or to clear multiple fractional powers. They often test the "degree undefined" cases rigorously.



Always double-check the definition of degree, especially the polynomial in derivatives condition. A common mistake is to state a degree when it should be undefined. Stay sharp!

🧠 Intuitive Understanding

Welcome to the intuitive understanding of Ordinary Differential Equations (ODEs), focusing on their order and degree. These concepts are fundamental to classifying and understanding how differential equations model real-world phenomena.



Intuitive Understanding of Order


Imagine you're describing how something changes over time or space. A differential equation does exactly that. The order of an ODE tells us about the highest level of 'change of change' involved in the relationship it describes.



  • What a derivative means: A first derivative ($frac{dy}{dx}$) represents the direct rate of change. For example, velocity is the rate of change of position.

  • What higher derivatives mean: A second derivative ($frac{d^2y}{dx^2}$) represents the rate of change of the rate of change. For instance, acceleration is the rate of change of velocity. A third derivative would be the rate of change of acceleration, and so on.

  • Intuition for Order: The order of an ODE is simply the highest number of times a variable has been differentiated in that equation. It indicates the "deepest" or "most complex" level of instantaneous change being considered.

    • A first-order ODE models processes where only the direct rate of change is relevant (e.g., population growth, radioactive decay, simple circuit analysis).

    • A second-order ODE models processes involving a rate of change of a rate of change (e.g., oscillatory motion, spring-mass systems, wave equations, dynamics).

    • Higher-order ODEs describe even more intricate relationships involving multiple layers of change.




In essence, the order tells you how many "layers" of differentiation you need to peel back to understand the instantaneous behavior described by the equation.



Intuitive Understanding of Degree


Once you've identified the highest order derivative in an equation, the degree tells you about the power to which that highest order derivative is raised. However, there's a crucial condition: the equation must be expressible as a polynomial in its derivatives.



  • Intuition for Degree: The degree represents the highest power of the highest order derivative, *after* the equation has been made free from radicals and fractional powers of derivatives, and is expressed as a polynomial in terms of its derivatives.

    • If the highest derivative appears simply as itself (power 1), the degree is 1. Such equations are often simpler and are sometimes called linear (with respect to the derivatives).

    • If the highest derivative is squared, cubed, or raised to some higher integer power, the degree corresponds to that power. This introduces non-linearity into the relationship described by the highest-level change.



  • When Degree is Undefined: The concept of degree is only meaningful if the differential equation can be written as a polynomial in its derivatives. If the highest derivative (or any derivative) appears inside a non-polynomial function like a trigonometric function ($sin(y')$), an exponential function ($e^{y''}$), or a logarithmic function ($log(y''')$), then the degree is considered undefined. This is analogous to saying that $sin(x)$ does not have a "degree" in the polynomial sense.






































Equation Highest Derivative Order (Intuition) Degree (Intuition)
$frac{dy}{dx} + y = x$ $frac{dy}{dx}$ 1 (Direct rate of change) 1 (Highest derivative is to power 1)
$frac{d^2y}{dx^2} + (frac{dy}{dx})^3 + y = 0$ $frac{d^2y}{dx^2}$ 2 (Rate of change of rate of change) 1 (Highest derivative is to power 1)
$(frac{d^3y}{dx^3})^2 + (frac{d^2y}{dx^2})^5 = 0$ $frac{d^3y}{dx^3}$ 3 (Three layers of change) 2 (Highest derivative is to power 2)
$sin(frac{dy}{dx}) + y = 0$ $frac{dy}{dx}$ 1 (Direct rate of change) Undefined (Highest derivative inside sin function)


Understanding order and degree is the very first step in analyzing differential equations, as they dictate the methods used for finding solutions.


JEE Main & CBSE Note: Both order and degree are fundamental concepts for both exams. JEE Main frequently tests the definition of degree, especially the condition for it to be defined (polynomial in derivatives, free from fractional powers/radicals).

🌍 Real World Applications

Real-World Applications of Ordinary Differential Equations (ODEs): Order and Degree



Ordinary Differential Equations (ODEs) are indispensable tools for modeling dynamic systems across various scientific and engineering disciplines. Understanding the order and degree of an ODE is not just a mathematical classification but provides crucial insights into the nature, complexity, and behavior of the real-world phenomena they represent.

The order of an ODE often directly corresponds to the number of independent physical quantities or state variables required to describe the system at any given moment, or the 'memory' a system has of its past. The degree, particularly its linearity or non-linearity, dictates how the system responds to inputs and influences the complexity of its analysis.



1. First-Order ODEs: Modeling Immediate Dependence


First-order ODEs involve only the first derivative of the dependent variable. They are used to model systems where the rate of change of a quantity depends solely on its current state.



  • Population Dynamics: The Malthusian growth model, $frac{dP}{dt} = kP$, describes how the rate of change of population ($P$) is proportional to the current population. This is a first-order, first-degree linear ODE.


  • Radioactive Decay: The rate of decay of a radioactive substance is proportional to the amount present, $frac{dN}{dt} = -lambda N$. This is another first-order, first-degree linear ODE, fundamental in nuclear physics and carbon dating.


  • Newton's Law of Cooling: $frac{dT}{dt} = -k(T - T_a)$, where $T$ is the object's temperature and $T_a$ is the ambient temperature. This first-order, first-degree linear ODE models how an object cools down or heats up to match its surroundings.


  • Chemical Reactions: Simple first-order reactions, where the rate depends on the concentration of one reactant, are modeled using first-order ODEs.



2. Second-Order ODEs: Incorporating Inertia and Oscillations


Second-order ODEs involve the second derivative, frequently representing acceleration or curvature. These are crucial for systems exhibiting inertia, restoring forces, or oscillatory behavior.



  • Newton's Second Law of Motion: The fundamental law $F = ma$ becomes a second-order ODE when force depends on position or velocity. For example, a simple spring-mass system: $mfrac{d^2x}{dt^2} + cfrac{dx}{dt} + kx = F(t)$, where $x$ is displacement. This is a second-order, first-degree linear ODE.


  • Pendulum Motion: For a simple pendulum, the equation of motion is $Lfrac{d^2 heta}{dt^2} + gsin heta = 0$. This is a second-order non-linear ODE due to the $sin heta$ term. For small angles, $sin heta approx heta$, making it a linear second-order ODE.


  • RLC Circuits: The current ($I$) or charge ($Q$) in an RLC series circuit is governed by second-order ODEs, such as $Lfrac{d^2Q}{dt^2} + Rfrac{dQ}{dt} + frac{1}{C}Q = E(t)$. This is a second-order, first-degree linear ODE.


  • Beam Deflection: In civil engineering, the deflection of beams under various loads is often described by a fourth-order ODE, $frac{d^4y}{dx^4} = frac{w(x)}{EI}$, where $y$ is deflection, $w(x)$ is load, $E$ is Young's modulus, and $I$ is moment of inertia. This is a fourth-order, first-degree linear ODE.



Significance for JEE and Beyond:


For JEE, while direct questions on applications of ODE order and degree are less common, understanding these connections strengthens your intuition for physics problems that rely on differential equations. Recognizing the order helps determine the number of initial conditions needed for a unique solution, a concept often tested in problem-solving. Furthermore, knowing if an ODE is linear (first-degree) or non-linear greatly influences the methods used for its solution and the behavior of the system it models.

Mastering ODEs is not just about solving them; it's about understanding their power to describe and predict the intricate dynamics of our universe.

🔄 Common Analogies
The concepts of order and degree of an Ordinary Differential Equation (ODE) are fundamental for understanding and classifying these equations. While straightforward, students sometimes confuse them, especially the 'degree' part when radicals or fractions are involved. Analogies can significantly aid in clarifying these definitions.

Common Analogies for Order and Degree of ODEs



Let's use a hierarchical structure analogy to understand these two crucial characteristics of a differential equation.

1. The "Management Hierarchy" Analogy for Order


Imagine a company where a complex problem needs to be solved. This problem represents our Ordinary Differential Equation. The solution to this problem is the unknown function `y`.

* The Staff: The unknown function `y` is like the entry-level staff member, involved in the daily operations.
* The Team Leaders (1st Derivative): The first derivative, `dy/dx` or `y'`, represents a team leader. They oversee the direct operations of the staff.
* The Department Heads (2nd Derivative): The second derivative, `d²y/dx²` or `y''`, represents a department head. They manage team leaders.
* The Senior Management (3rd Derivative and higher): `d³y/dx³` or `y'''` and beyond represent progressively higher levels of management – VPs, CEOs, etc. – each overseeing the level below them.

The Order of the Differential Equation is determined by the highest level of management involved in the problem.

* If the problem primarily involves staff and team leaders (`y` and `y'`), it's a "team leader level" problem – First Order.
* If a department head is the highest-ranking official involved (`y''`), it's a "department head level" problem – Second Order.
* Crucial Point: Even if a team leader (`y'`) is shouting very loudly (i.e., has a high power like `(y')⁵`), if a department head (`y''`) is also present, the problem is still a "department head level" problem. The order is determined *only* by the rank of the highest derivative, not its power.

2. The "Influence/Authority" Analogy for Degree


Once you've identified the highest-ranking official involved (i.e., the order of the ODE), the Degree of the Differential Equation represents the power or influence of *that specific highest-ranking official* in the overall equation, *after ensuring all bureaucratic hurdles are cleared*.

* Let's say a department head (`d²y/dx²`) is the highest-ranking official present.
* If `(d²y/dx²)¹` appears, that department head has a standard level of influence – Degree 1.
* If `(d²y/dx²)³` appears, that department head has three times the influence or authority in the decision-making – Degree 3.
* The "Bureaucratic Hurdle" (JEE Specific): Before assessing the department head's influence (degree), all reports must be "clean" and "standardized." This means the differential equation must be made free of any radicals (like square roots `√`) or fractional powers (`( )¹/²`) involving the derivatives. You cannot assess the true influence if their authority is hidden or fractional.
* For example, if you have `√(d²y/dx²) + dy/dx = 0`, you must first remove the radical by isolating the radical term and squaring both sides: `d²y/dx² = (-dy/dx)²`. Now, the highest derivative is `d²y/dx²`, and its power is 1. So, the degree is 1.
* This rationalization step is often tested in JEE Main, making it a key differentiator from simpler CBSE problems that usually present equations already in rational polynomial form with respect to derivatives.

By using these analogies, you can conceptually distinguish between order (the highest "rank" of derivative) and degree (the "power" of that highest-ranked derivative after simplification), making it easier to tackle related problems.
📋 Prerequisites

To effectively grasp the concepts of Ordinary Differential Equations (ODEs), especially their order and degree, a solid foundation in certain fundamental mathematical topics is essential. These prerequisites ensure that you can understand the building blocks of differential equations and correctly identify their defining characteristics.



Here are the key concepts you should be comfortable with before diving into ODEs, order, and degree:




  • Functions and Variables:

    • Understanding of a function, typically represented as y = f(x), where x is the independent variable and y is the dependent variable. This distinction is fundamental as differential equations relate a function to its derivatives with respect to one or more independent variables. For ODEs, we are specifically dealing with one independent variable.

    • Familiarity with various types of functions (polynomial, trigonometric, exponential, logarithmic) and their properties.



  • Differential Calculus (Core Prerequisite):

    • Concept of Derivative: A clear understanding of what a derivative represents (rate of change, slope of tangent).

    • Standard Differentiation Formulas: Proficiency in differentiating elementary functions (e.g., d/dx (x^n), d/dx (sin x), d/dx (e^x), d/dx (log x), etc.).

    • Rules of Differentiation: Mastery of the product rule, quotient rule, and chain rule. These are frequently used when forming or manipulating expressions within differential equations.

    • Higher-Order Derivatives: This is critically important for understanding the order of a differential equation. You must know how to calculate second derivatives (d²y/dx²), third derivatives (d³y/dx³), and so on. The highest order of the derivative present in an equation directly defines the order of the differential equation.

    • Implicit Differentiation: Ability to differentiate implicit functions, as differential equations often involve expressions where variables are not explicitly separated.



  • Algebraic Manipulation:

    • Polynomials and their Degree: Understanding the concept of the degree of a polynomial (the highest power of the variable). This directly translates to determining the degree of a differential equation, which is the highest power of the highest-order derivative after clearing all fractional and radical powers.

    • Solving Equations: Basic skills in solving algebraic equations and manipulating expressions to isolate terms or simplify.

    • Exponents and Radicals: Familiarity with rules of exponents and how to deal with square roots, cube roots, etc., as these may appear in differential equations and need to be cleared to find the degree.





JEE Main vs. CBSE Board Exams Callout:

Both JEE Main and CBSE Board exams expect a strong grasp of these fundamental calculus and algebra concepts. For JEE, the application and speed in differentiation and algebraic manipulation are more rigorously tested. For determining order and degree, understanding higher-order derivatives and the degree of polynomials (after clearing fractions/radicals) is equally vital for both.



Ensure you are confident in these topics before proceeding. A quick review of your Class 11 and 12 calculus notes will be highly beneficial.

🔗 Related Topics

Understanding "Ordinary Differential Equations (ODEs): Order and Degree" is foundational for tackling the broader topic of Differential Equations in JEE Main. This section outlines the topics that are either prerequisites to this concept or are direct extensions that build upon this fundamental understanding.



Mastering these related areas will provide a holistic view and prepare you for solving complex problems involving differential equations.



Related Topics to ODEs, Order, and Degree




  • Calculus Fundamentals (Differentiation and Integration):

    • Differentiation: The concept of derivatives is central to differential equations. An ODE is, by definition, an equation involving derivatives of an unknown function. Understanding higher-order derivatives is crucial for determining the order of a differential equation.

    • Integration: Solving differential equations invariably involves integration. Techniques of indefinite and definite integration, along with understanding constants of integration, are indispensable for finding general and particular solutions.




  • Families of Curves:

    • Understanding how a single equation can represent a family of curves (e.g., $y = mx + c$ for a family of straight lines) is critical. Differential equations often represent the differential relation common to a family of curves, which leads directly to the concept of formation of differential equations.




  • Formation of Differential Equations:

    • This is a direct application of knowing what a differential equation is and how its order relates to the number of arbitrary constants removed. Given a family of curves, you are expected to form the differential equation by eliminating arbitrary constants. The order of the resulting differential equation will be equal to the number of arbitrary constants.




  • Methods of Solving First Order, First Degree Differential Equations:

    • Once you can identify the order and degree, the next logical step is to learn how to solve them. For JEE Main, the primary focus is on first-order and first-degree ODEs. Key methods include:

      • Variable Separable Method

      • Homogeneous Differential Equations and equations reducible to homogeneous form.

      • Linear Differential Equations of the form $frac{dy}{dx} + Py = Q$ or $frac{dx}{dy} + Px = Q$.



    • These methods directly rely on the equation being of a specific order and degree.




  • Initial Value Problems (IVPs) and Boundary Value Problems (BVPs):

    • After finding the general solution of a differential equation, initial conditions (e.g., $y(x_0) = y_0$) or boundary conditions are used to find the specific value(s) of the arbitrary constant(s), leading to a particular solution. This is a common exam question format.

    • JEE Main Insight: Most problems in JEE Main are IVPs, providing a single point to determine the constant of integration.




  • Applications of Differential Equations:

    • Differential equations model real-world phenomena across various fields. Understanding order and degree helps in classifying these models. Common applications relevant to JEE include:

      • Population growth/decay models

      • Radioactive decay

      • Newton's Law of Cooling

      • Basic physics problems (e.g., motion with air resistance)



    • These applications often involve forming and then solving first-order ODEs.





By building a strong foundation in these related areas, you will be well-equipped to handle the differential equations section in your exams.

⚠️ Common Exam Traps

Common Exam Traps: Order and Degree of ODEs


Understanding the order and degree of a differential equation is fundamental, yet students frequently fall into specific traps during exams. Being aware of these pitfalls can significantly improve accuracy, especially in competitive exams like JEE Main.



Traps Related to Determining Order



  • Confusing notation: While primarily dealing with ODEs, sometimes notation can be ambiguous or deliberately tricky. Always identify the highest order *ordinary* derivative present in the equation. For example, $(dy/dx)$ is first order, $(d^2y/dx^2)$ is second order, etc.

  • Premature simplification: Ensure you've identified the highest order derivative after the equation is fully written out, not during an intermediate step of simplification if that simplification reduces the apparent order.



Major Traps Related to Determining Degree


The degree is far more prone to traps than the order. Remember the definition: the degree is the power of the highest order derivative, *after* the equation has been made free from radicals and fractions as far as the derivatives are concerned, and is expressible as a polynomial in derivatives.




  • Equation not a polynomial in derivatives: This is the most significant trap, particularly in JEE.

    • If the differential equation contains derivatives within transcendental functions (like trigonometric, logarithmic, exponential functions), its degree is undefined.

    • Examples:

      • $e^{dy/dx} + x = 0$

      • $sin(d^2y/dx^2) = x+y$

      • $log(d^3y/dx^3) = cos x$

      • $sqrt{1 + (dy/dx)^2} = d^2y/dx^2$ (after squaring, it becomes polynomial, but be careful with powers like $e^{sqrt{dy/dx}}$)




    JEE Specific Callout: Questions with 'degree undefined' are very common in JEE Main. Always check this condition first before trying to find the degree.


    CBSE Callout: CBSE board exams generally provide equations where the degree is defined, requiring students to clear radicals/fractions.




  • Presence of Radicals/Fractional Powers of Derivatives: Before determining the degree, the differential equation *must* be rationalized and cleared of all fractional powers/radicals involving any derivative.

    • Trap: Students often directly pick the power without clearing radicals. For example, in $(1 + (dy/dx)^2)^{3/2} = d^2y/dx^2$, if one doesn't square both sides, they might mistakenly say degree is $1$ (from $d^2y/dx^2$) or $3/2$ (which is invalid).

    • Correct approach: To clear $(1 + (dy/dx)^2)^{3/2} = d^2y/dx^2$, cube both sides first to get $(1 + (dy/dx)^2)^3 = (d^2y/dx^2)^2$. Only now identify the highest order derivative ($d^2y/dx^2$) and its power ($2$). So, Order = 2, Degree = 2.




  • Negative Powers of Derivatives: Ensure all derivative terms have non-negative integer powers. Clear any denominators involving derivatives (e.g., $1/(dy/dx)$ terms).


  • Misidentifying the Power: After clearing radicals and ensuring it's a polynomial in derivatives, the degree is the power of the *highest order derivative*, not necessarily the highest power present in the entire equation.

    • Trap: In $(d^2y/dx^2)^2 + (dy/dx)^5 + y = 0$, the highest order derivative is $d^2y/dx^2$. Its power is 2. The degree is 2, not 5.





Illustrative Example of a Common Trap


Consider the differential equation: $(1 + (frac{dy}{dx})^2)^{3/2} = k frac{d^2y}{dx^2}$


Common Mistakes:



  1. Stating degree as $3/2$ (fractional power, invalid).

  2. Stating degree as $1$ (power of $d^2y/dx^2$ before squaring).


Correct Approach:



  1. To remove the fractional power, square both sides:
    $(1 + (frac{dy}{dx})^2)^3 = k^2 (frac{d^2y}{dx^2})^2$

  2. The equation is now a polynomial in derivatives.

  3. Highest order derivative: $frac{d^2y}{dx^2}$ (Order = 2).

  4. Power of the highest order derivative: $2$ (Degree = 2).



Stay vigilant and systematically check for these conditions to avoid losing marks on what seem like basic questions. Practice with a variety of challenging examples will solidify your understanding.


⭐ Key Takeaways

Key Takeaways: Order and Degree of ODEs



Understanding the order and degree of a differential equation is fundamental for its classification and subsequent solution methods. These concepts are frequently tested in both CBSE Board exams and JEE Main.



1. What is an Ordinary Differential Equation (ODE)?



  • An equation involving an independent variable (e.g., x), a dependent variable (e.g., y), and the derivatives of the dependent variable with respect to the independent variable (e.g., dy/dx, d2y/dx2).

  • It contains derivatives with respect to only one independent variable.



2. Order of a Differential Equation



  • The order of a differential equation is the order of the highest derivative appearing in the equation.

  • It is always a positive integer.

  • Example: For the equation (d2y/dx2) + x(dy/dx)3 = sin(x), the highest derivative is d2y/dx2, which is a second-order derivative. Hence, the order of the ODE is 2.

  • JEE/CBSE Note: The order is always defined for any differential equation.



3. Degree of a Differential Equation



  • The degree of a differential equation is the power of the highest order derivative, provided that the differential equation can be expressed as a polynomial in derivatives.

  • Crucial Condition: Before determining the degree, the differential equation must be made free from radicals (fractional powers) and fractions involving derivatives. It must be a polynomial in all its derivatives.

  • It is always a positive integer.

  • JEE Main Specific: Cases where Degree is Undefined

    • If the differential equation cannot be expressed as a polynomial in its derivatives. This occurs when derivatives are involved in transcendental functions (e.g., trigonometric functions like sin(y'), cos(y''), exponential functions like ey', logarithmic functions like log(y'''')).

    • Example:

      • sin(dy/dx) + y = 0 – Degree is undefined.

      • e(d2y/dx2) + x = 0 – Degree is undefined.





  • Example for defined degree: Consider (1 + (dy/dx)2)3/2 = d2y/dx2.

    1. To find the degree, first clear the fractional power: Square both sides: (1 + (dy/dx)2)3 = (d2y/dx2)2.

    2. The highest order derivative is d2y/dx2.

    3. Its power in the polynomial form is 2. Thus, the degree is 2. (Order is also 2).





4. Key Differentiator (CBSE vs. JEE Main)



























Feature Order Degree
Definition Highest derivative order Power of highest order derivative (after making it polynomial in derivatives)
Always Defined? Yes Not always (Undefined if not a polynomial in derivatives)
Value Type Positive integer Positive integer (if defined)


Mastering these definitions is crucial as they are foundational for further topics in differential equations. Always remember to check for fractional powers and transcendental functions of derivatives before concluding the degree!


🧩 Problem Solving Approach

Problem Solving Approach: Order and Degree of Differential Equations


Determining the order and degree of a differential equation is a fundamental skill. While the concept might seem straightforward, certain forms of equations require careful steps to correctly identify the degree. Follow this systematic approach:



Step 1: Identify the Order of the Differential Equation



  1. Scan the Equation: Look for all derivative terms present (e.g., $frac{dy}{dx}$, $frac{d^2y}{dx^2}$, $frac{d^3y}{dx^3}$, etc.).

  2. Determine the Highest Order Derivative: The order of the highest derivative present in the differential equation is its order.

    • For example, if $frac{d^3y}{dx^3}$ is the highest derivative, the order is 3.



  3. JEE Tip: The order is always a positive integer. It's usually very direct to find.



Step 2: Prepare the Equation for Determining the Degree


The degree of a differential equation is defined only when it can be expressed as a polynomial in its derivatives. This means:



  1. Eliminate Radicals and Fractional Powers of Derivatives: If any derivative term (e.g., $frac{dy}{dx}$, $frac{d^2y}{dx^2}$) is under a radical sign or raised to a fractional power, you must clear these first.

    • Raise both sides of the equation to an appropriate power to remove all fractional exponents/radicals involving derivatives.

    • Example: If you have $(frac{d^2y}{dx^2})^{frac{3}{2}}$, square both sides. If you have $sqrt{frac{dy}{dx}}$, square both sides.



  2. Check for Transcendental Functions of Derivatives: Examine if any derivative term is an argument of a transcendental function (e.g., $sin(frac{dy}{dx})$, $cos(frac{d^2y}{dx^2})$, $e^{frac{dy}{dx}}$, $log(frac{d^3y}{dx^3})$).

    • If such terms are present, the differential equation cannot be expressed as a polynomial in its derivatives. In this case, the degree is undefined.

    • Warning: Be extremely careful with this step, as it's a common trap in JEE problems.





Step 3: Determine the Degree of the Differential Equation



  1. Once the Equation is a Polynomial in Derivatives: After ensuring the equation is free from fractional powers/radicals of derivatives and transcendental functions of derivatives, identify the highest order derivative again (from Step 1).

  2. Find the Power: The power (exponent) of this highest order derivative, after the equation has been rationalized and made polynomial in derivatives, is its degree.

    • For example, if the equation is $( frac{d^2y}{dx^2} )^3 + 5 (frac{dy}{dx})^2 + y = 0$, the highest order derivative is $frac{d^2y}{dx^2}$, and its power is 3. So, the degree is 3.



  3. CBSE vs. JEE: CBSE questions are generally direct. JEE questions often involve clearing fractional powers/radicals or checking for undefined degrees.



Illustrative Example:


Determine the order and degree of the differential equation: $(1 + (frac{dy}{dx})^2)^{3/2} = k frac{d^2y}{dx^2}$



  • Step 1: Identify Order

    • The derivatives present are $frac{dy}{dx}$ and $frac{d^2y}{dx^2}$.

    • The highest order derivative is $frac{d^2y}{dx^2}$.

    • Therefore, the Order = 2.



  • Step 2: Prepare for Degree (Clear Fractional Powers)

    • The term $(1 + (frac{dy}{dx})^2)^{3/2}$ has a fractional power of $3/2$. To remove it, square both sides of the equation:

    • $[ (1 + (frac{dy}{dx})^2)^{3/2} ]^2 = [ k frac{d^2y}{dx^2} ]^2$

    • $(1 + (frac{dy}{dx})^2)^3 = k^2 (frac{d^2y}{dx^2})^2$

    • Now, the equation is a polynomial in derivatives. There are no radicals or transcendental functions of derivatives.



  • Step 3: Determine Degree

    • The highest order derivative is $frac{d^2y}{dx^2}$ (from the right side of the simplified equation).

    • Its power (exponent) in the simplified equation is 2.

    • Therefore, the Degree = 2.




📝 CBSE Focus Areas

CBSE Focus Areas: Order and Degree of Differential Equations



Understanding the order and degree of ordinary differential equations (ODEs) is a fundamental concept frequently tested in CBSE Board Examinations. While seemingly straightforward, certain nuances, especially concerning the degree, are common areas where students make mistakes. CBSE questions typically focus on direct application of definitions and careful handling of equations that require simplification before determining the degree.


1. Order of a Differential Equation:



The order of a differential equation is defined as the order of the highest derivative appearing in the equation. It is always a positive integer.



  • CBSE Relevance: Determining the order is generally simpler and less prone to error. Simply identify the highest derivative (e.g., $dy/dx$, $d^2y/dx^2$, $d^3y/dx^3$, etc.) and its order.

  • Example: In the equation $frac{d^3y}{dx^3} + 5x left(frac{dy}{dx}
    ight)^2 - 7y = 0$, the highest derivative is $frac{d^3y}{dx^3}$, so the order is 3.




2. Degree of a Differential Equation:



The degree of a differential equation is the power of the highest order derivative, provided that the differential equation can be expressed as a polynomial in its derivatives.



  • CBSE Relevance (Crucial Points):

    • Polynomial in Derivatives: For the degree to be defined, the differential equation must be expressible as a polynomial in its derivatives. This means derivatives should not appear inside trigonometric functions, logarithmic functions, exponential functions, etc.

    • Free from Radicals/Fractional Powers: The equation must be free from radicals or fractional powers of derivatives. If such powers exist, they must first be cleared by suitable algebraic manipulations (e.g., squaring both sides).

    • Degree Not Defined: If a differential equation cannot be expressed as a polynomial in its derivatives (e.g., $sin(dy/dx)$, $e^{d^2y/dx^2}$, $log(dy/dx)$), then its degree is not defined.






3. Key Points for CBSE Examinations:



To confidently tackle CBSE questions on order and degree, remember these practical tips:



  • Always simplify first: If there are radicals or fractional powers involving derivatives, eliminate them algebraically. For example, $(1 + (dy/dx)^2)^{3/2} = d^2y/dx^2$ would require squaring both sides to get $(1 + (dy/dx)^2)^3 = (d^2y/dx^2)^2$.

  • Check for transcendental functions: If derivatives are arguments of transcendental functions (like $sin(dy/dx)$ or $e^{d^2y/dx^2}$), the degree is not defined. The order, however, will still be defined.

  • Identify highest order: First, correctly identify the highest order derivative in the simplified equation.

  • Determine its power: Then, look at the power of *that specific highest order derivative* to find the degree.




Example for CBSE:



Find the order and degree of the differential equation: $(y'')^2 + (y')^3 + y = e^x + sin(x)$



  1. Highest Order Derivative: The derivatives present are $y''$ and $y'$. The highest order derivative is $y''$ (which is $d^2y/dx^2$).

  2. Order: Since the highest order derivative is $y''$, the order of the differential equation is 2.

  3. Polynomial in Derivatives Check: The equation $(y'')^2 + (y')^3 + y = e^x + sin(x)$ is a polynomial in its derivatives $y''$ and $y'$. There are no fractional powers of derivatives, and derivatives are not inside transcendental functions.

  4. Power of Highest Order Derivative: The power of the highest order derivative ($y''$) is 2.

  5. Degree: Therefore, the degree of the differential equation is 2.




Remember to always check both the definition of order and the specific conditions for degree carefully, especially the "polynomial in derivatives" aspect, to secure full marks in CBSE.

🎓 JEE Focus Areas

JEE Focus Areas: Order and Degree of Differential Equations



Understanding the order and degree of a differential equation is fundamental, particularly for classifying and solving them. While seemingly simple, JEE Main often tests these concepts with subtle twists, especially concerning the degree.

1. Order of a Differential Equation



The order of a differential equation is defined as the order of the highest derivative present in the equation. It's relatively straightforward to determine.


  • Key Point: The order is always a positive integer.

  • JEE Tip: Focus on identifying the derivative with the maximum number of prime marks or the highest $n$ in $frac{d^n y}{dx^n}$.



2. Degree of a Differential Equation



The degree of a differential equation is defined as the power of the highest order derivative when the differential equation is made free from radicals and fractional powers with respect to all the derivatives.


  • Crucial Conditions for Degree:

    1. The differential equation must be expressible as a polynomial in all its derivatives ($y', y'', y'''$, etc.). This means there should be no terms like $sin(y')$, $log(y'')$, $e^{y'}$, etc.

    2. All radicals and fractional powers involving derivatives must be cleared by appropriate algebraic manipulation (e.g., squaring both sides).



  • JEE Trap: If a differential equation cannot be expressed as a polynomial in its derivatives (e.g., contains $sin(y')$, $cos(y'')$, $e^{y'}$, $log(y''')$), its degree is undefined. Its order, however, can still be determined.

  • CBSE vs. JEE: CBSE often presents simpler equations. JEE Main frequently includes equations with fractional powers or non-polynomial derivative terms to test a deeper understanding of the degree definition.



3. Steps to Determine Order and Degree for JEE Problems




  1. Identify all derivatives: List $y', y'', y'''$, etc., present in the equation.

  2. Determine the Highest Order Derivative: The order of this derivative is the order of the differential equation.

  3. Check for Polynomial Form: Ensure the equation is a polynomial in its derivatives. If not (e.g., $sin(y')$ term), the degree is undefined.

  4. Clear Radicals/Fractional Powers: If the equation contains terms like $(y'')^{3/2}$ or $sqrt{y'}$, manipulate the equation algebraically (e.g., by squaring, cubing) until all powers of derivatives are integers.

  5. Find Power of Highest Order Derivative: After all manipulations, the power of the highest order derivative (identified in step 2) is the degree.



Example for JEE Main:



Consider the differential equation: $(1 + (frac{dy}{dx})^2)^{3/2} = k frac{d^2y}{dx^2}$


  1. Highest Order Derivative: $frac{d^2y}{dx^2}$ (or $y''$).

  2. Order: 2.

  3. Clear Fractional Power: To clear the $3/2$ power, square both sides:
    $(1 + (frac{dy}{dx})^2)^3 = (k frac{d^2y}{dx^2})^2$
    $(1 + (y')^2)^3 = k^2 (y'')^2$

  4. Check Polynomial Form: Yes, it is a polynomial in $y'$ and $y''$.

  5. Power of Highest Order Derivative: The highest order derivative is $y''$, and its power in the cleared equation is 2.

  6. Degree: 2.




Summary Table: Order vs. Degree
































Feature Order Degree
Definition Order of highest derivative. Power of highest derivative after clearing radicals/fractions and expressing as polynomial in derivatives.
Always Defined? Yes, for any ODE. No, undefined if not a polynomial in derivatives (e.g., $sin(y')$).
Value Type Positive integer. Positive integer (if defined).
JEE Focus Direct identification. Algebraic manipulation, checking for polynomial form, undefined cases.



Mastering these distinctions will save valuable marks in JEE Main. Always be cautious with fractional powers and non-polynomial functions of derivatives!

📊 AI Generation Metadata

AI Model: Gemini Full Parallel API Client
Status: AI Generated
Version: 1
Generated: Oct 22, 2025
🌐 Overview
An ordinary differential equation (ODE) relates a function and its derivatives with respect to a single independent variable. Order is the highest derivative order present. Degree is the power of the highest-order derivative when the equation is polynomial in derivatives (after clearing radicals/fractions of derivatives).
📚 Fundamentals
• Order: max index in d^n y/dx^n present.
• Degree: exponent of highest derivative after clearing radicals/fractions (if possible).
• Not defined if equation is not polynomial in derivatives (e.g., sin(y′) = x).
🔬 Deep Dive
Why degree is less central in modern ODE theory; relation to polynomial differential equations and solvability classes (overview).
🎯 Shortcuts
“Order is how high; degree is how powered (if algebraic).”
💡 Quick Tips
• y′′ + (y′)^2 = 0 → order 2, degree 1 (power of y′′).
• [y′]^{1/2} + y = x → degree undefined until squared.
• sin(y′) = x → degree undefined (non-polynomial).
🧠 Intuitive Understanding
Order tells “how many layers of rates” are involved; degree tells the highest power to which that topmost rate appears (provided the equation is polynomial in derivatives).
🌍 Real World Applications
• Motion (2nd order for acceleration), circuits (1st/2nd order), population dynamics (1st order). Understanding order/degree guides solution methods.
🔄 Common Analogies
• Order like “how many nested derivatives”; degree like “exponent of the top derivative” if the equation is algebraic in derivatives.
📋 Prerequisites
Derivatives and notation; implicit/explicit equations; algebraic manipulation to clear roots/fractions of derivatives if needed to define degree.
🔗 Related Topics
Linear vs nonlinear ODEs; order reduction; initial vs boundary value problems; existence/uniqueness (awareness).
⚠️ Common Exam Traps
• Reporting degree without clearing radicals/fractions first.
• Confusing order with degree.
• Declaring degree for non-polynomial-in-derivatives equations.
⭐ Key Takeaways
• Order: highest derivative order.
• Degree: power of that derivative when algebraic in derivatives.
• Some ODEs have undefined degree (non-polynomial in derivatives).
🧩 Problem Solving Approach
1) Locate the highest derivative present.
2) Multiply through to remove denominator/radical of derivatives.
3) Check algebraic in derivatives; if yes, read degree; if not, state “degree undefined.”
📝 CBSE Focus Areas
Correct identification of order and degree in standard examples; recognizing when degree is undefined.
🎓 JEE Focus Areas
Tricky forms needing algebraic cleanup; implicit equations with mixed derivatives; examples vs counterexamples.

📊 AI Generation Metadata

Estimated Time: 45 mins
AI Model: ChatGPT 5 (Copilot Agent)
Status: AI Generated
Version: 1
Generated: Oct 23, 2025

📝CBSE 12th Board Problems (18)

Problem 255
Medium 1 Mark
Find the order and degree of the differential equation: sin(d^2y/dx^2) + dy/dx + y = 0
Show Solution
1. Identify the highest order derivative, which is d^2y/dx^2.<br>2. The order is 2.<br>3. Observe that the highest order derivative (d^2y/dx^2) is inside a trigonometric function (sin).<br>4. This means the equation cannot be expressed as a polynomial in its derivatives.<br>5. Therefore, the degree is undefined.
Final Answer: Order = 2, Degree = Undefined
Problem 255
Hard 2 Marks
Find the order and degree of the differential equation: <span style='color: #0000FF;'>x(d<sup>2</sup>y/dx<sup>2</sup>) + (dy/dx)<sup>2</sup> - y(dy/dx) = 0</span>
Show Solution
<ul><li>Identify the highest order derivative in the equation. The highest order derivative is <span style='color: #0000FF;'>d<sup>2</sup>y/dx<sup>2</sup></span>. So, the <strong>order is 2</strong>.</li><li>Check if the equation is a polynomial in its derivatives. All terms involve derivatives raised to integer powers, and there are no derivatives inside transcendental functions or radicals. Therefore, the equation is already a polynomial in its derivatives.</li><li>Identify the highest power of the highest order derivative (<span style='color: #0000FF;'>d<sup>2</sup>y/dx<sup>2</sup></span>). The term is <span style='color: #0000FF;'>x(d<sup>2</sup>y/dx<sup>2</sup>)</span>, and the power of <span style='color: #0000FF;'>d<sup>2</sup>y/dx<sup>2</sup></span> is <strong>1</strong>.</li></ul>
Final Answer: Order = 2, Degree = 1
Problem 255
Hard 1 Mark
Determine the order and degree of the differential equation: <span style='color: #0000FF;'>y'' + 3y' + 2y = log(x)</span>
Show Solution
<ul><li>Identify the highest order derivative. Here, <span style='color: #0000FF;'>y''</span> represents <span style='color: #0000FF;'>d<sup>2</sup>y/dx<sup>2</sup></span>, which is a second-order derivative. <span style='color: #0000FF;'>y'</span> represents <span style='color: #0000FF;'>dy/dx</span>, a first-order derivative.</li><li>The highest order derivative is <span style='color: #0000FF;'>y''</span>. So, the <strong>order is 2</strong>.</li><li>The equation is already in a polynomial form in terms of its derivatives.</li><li>Identify the highest power of the highest order derivative (<span style='color: #0000FF;'>y''</span>). The term is <span style='color: #0000FF;'>y''</span>, which has a power of <strong>1</strong>.</li></ul>
Final Answer: Order = 2, Degree = 1
Problem 255
Hard 2 Marks
Find the order and degree of the differential equation: <span style='color: #0000FF;'>&radic;[1 + (dy/dx)<sup>2</sup>] = k(d<sup>2</sup>y/dx<sup>2</sup>)</span>
Show Solution
<ul><li>The presence of the square root indicates that the equation is not yet in polynomial form concerning derivatives.</li><li>Square both sides of the equation to eliminate the radical: <span style='color: #0000FF;'>[1 + (dy/dx)<sup>2</sup>] = [k(d<sup>2</sup>y/dx<sup>2</sup>)]<sup>2</sup></span></li><li>Simplify: <span style='color: #0000FF;'>1 + (dy/dx)<sup>2</sup> = k<sup>2</sup>(d<sup>2</sup>y/dx<sup>2</sup>)<sup>2</sup></span></li><li>The equation is now a polynomial in its derivatives.</li><li>Identify the highest order derivative: <span style='color: #0000FF;'>d<sup>2</sup>y/dx<sup>2</sup></span>. Thus, the <strong>order is 2</strong>.</li><li>Identify the highest power of the highest order derivative: The term <span style='color: #0000FF;'>k<sup>2</sup>(d<sup>2</sup>y/dx<sup>2</sup>)<sup>2</sup></span> has the highest power of the highest order derivative, which is <strong>2</strong>.</li></ul>
Final Answer: Order = 2, Degree = 2
Problem 255
Hard 1 Mark
Determine the order and degree of the differential equation: <span style='color: #0000FF;'>sin(dy/dx) + e<sup>(d<sup>2</sup>y/dx<sup>2</sup>)</sup> = x<sup>2</sup></span>
Show Solution
<ul><li>Identify the highest order derivative present in the equation. Here, it is <span style='color: #0000FF;'>d<sup>2</sup>y/dx<sup>2</sup></span>. So, the <strong>order is 2</strong>.</li><li>Check if the differential equation can be expressed as a polynomial in terms of its derivatives.</li><li>The terms <span style='color: #0000FF;'>sin(dy/dx)</span> and <span style='color: #0000FF;'>e<sup>(d<sup>2</sup>y/dx<sup>2</sup>)</sup></span> involve derivatives as arguments of non-polynomial functions (trigonometric and exponential functions).</li><li>Therefore, the equation cannot be written as a polynomial in its derivatives.</li></ul>
Final Answer: Order = 2, Degree = Not Defined
Problem 255
Hard 2 Marks
Find the order and degree of the differential equation: <span style='color: #0000FF;'>(d<sup>2</sup>y/dx<sup>2</sup>)<sup>3/2</sup> = x + (dy/dx)</span>
Show Solution
<ul><li>To remove the fractional power, square both sides of the equation: <span style='color: #0000FF;'>[(d<sup>2</sup>y/dx<sup>2</sup>)<sup>3/2</sup>]<sup>2</sup> = [x + (dy/dx)]<sup>2</sup></span></li><li>This simplifies to: <span style='color: #0000FF;'>(d<sup>2</sup>y/dx<sup>2</sup>)<sup>3</sup> = (x + dy/dx)<sup>2</sup></span></li><li>Expand the right side if necessary, but it's already a polynomial in derivatives: <span style='color: #0000FF;'>(d<sup>2</sup>y/dx<sup>2</sup>)<sup>3</sup> = x<sup>2</sup> + 2x(dy/dx) + (dy/dx)<sup>2</sup></span></li><li>Identify the highest order derivative: <span style='color: #0000FF;'>d<sup>2</sup>y/dx<sup>2</sup></span>. This is a <strong>second-order</strong> derivative.</li><li>Identify the highest power of the highest order derivative: The term <span style='color: #0000FF;'>(d<sup>2</sup>y/dx<sup>2</sup>)<sup>3</sup></span> has the highest power, which is <strong>3</strong>.</li></ul>
Final Answer: Order = 2, Degree = 3
Problem 255
Hard 2 Marks
Determine the order and degree of the following differential equation: <span style='color: #0000FF;'>y = x(dy/dx) + &radic;[1 + (dy/dx)<sup>2</sup>]</span>
Show Solution
<ul><li>Rearrange the equation to isolate the radical term: <span style='color: #0000FF;'>y - x(dy/dx) = &radic;[1 + (dy/dx)<sup>2</sup>]</span></li><li>Square both sides to eliminate the radical: <span style='color: #0000FF;'>(y - x(dy/dx))<sup>2</sup> = 1 + (dy/dx)<sup>2</sup></span></li><li>Expand the left side: <span style='color: #0000FF;'>y<sup>2</sup> - 2xy(dy/dx) + x<sup>2</sup>(dy/dx)<sup>2</sup> = 1 + (dy/dx)<sup>2</sup></span></li><li>Rearrange into a polynomial in terms of derivatives: <span style='color: #0000FF;'>(x<sup>2</sup> - 1)(dy/dx)<sup>2</sup> - 2xy(dy/dx) + (y<sup>2</sup> - 1) = 0</span></li><li>Identify the highest order derivative: <span style='color: #0000FF;'>dy/dx</span>, which is a <strong>first-order</strong> derivative.</li><li>Identify the highest power of the highest order derivative: The term <span style='color: #0000FF;'>(dy/dx)<sup>2</sup></span> has the highest power, which is <strong>2</strong>.</li></ul>
Final Answer: Order = 1, Degree = 2
Problem 255
Medium 2 Marks
Find the order and degree of the differential equation: [1 + (dy/dx)^2]^(3/2) = d^2y/dx^2
Show Solution
1. To remove the fractional power, square both sides of the equation: ([1 + (dy/dx)^2]^(3/2))^2 = (d^2y/dx^2)^2.<br>2. This simplifies to [1 + (dy/dx)^2]^3 = (d^2y/dx^2)^2.<br>3. The equation is now free of radicals and fractional powers and is a polynomial in its derivatives.<br>4. Identify the highest order derivative, which is d^2y/dx^2.<br>5. The order is 2.<br>6. The power of the highest order derivative (d^2y/dx^2) is 2.<br>7. Therefore, the degree is 2.
Final Answer: Order = 2, Degree = 2
Problem 255
Medium 1 Mark
State the order and degree of the differential equation: e^(dy/dx) + x = 0
Show Solution
1. Identify the highest order derivative, which is dy/dx.<br>2. The order is 1.<br>3. Observe that the highest order derivative (dy/dx) is in the exponent of an exponential function (e).<br>4. This means the equation cannot be expressed as a polynomial in its derivatives.<br>5. Therefore, the degree is undefined.
Final Answer: Order = 1, Degree = Undefined
Problem 255
Easy 1 Mark
Find the order and degree of the differential equation: (d^2y/dx^2)^3 + (dy/dx)^2 + y = x
Show Solution
1. Identify the highest order derivative in the equation. In this case, it is d^2y/dx^2. 2. The order of the differential equation is the order of the highest derivative, which is 2. 3. The degree of the differential equation is the power of the highest order derivative, provided the equation is a polynomial in its derivatives. Here, the power of d^2y/dx^2 is 3. 4. Since the equation is a polynomial in its derivatives, the degree is 3.
Final Answer: Order = 2, Degree = 3
Problem 255
Medium 1 Mark
What is the order and degree of the differential equation: (d^3y/dx^3)^(2/3) + d^2y/dx^2 + y = 0?
Show Solution
1. To remove the fractional power, isolate the term with the fractional power and then cube both sides: (d^3y/dx^3)^(2/3) = -(d^2y/dx^2 + y).<br>2. Cubing both sides gives: (d^3y/dx^3)^2 = -(d^2y/dx^2 + y)^3.<br>3. This equation is now a polynomial in its derivatives.<br>4. The highest order derivative is d^3y/dx^3, so the order is 3.<br>5. The power of the highest order derivative (d^3y/dx^3) is 2.<br>6. Therefore, the degree is 2.
Final Answer: Order = 3, Degree = 2
Problem 255
Medium 1 Mark
Determine the order and degree of the differential equation: dy/dx = sqrt(x + d^2y/dx^2)
Show Solution
1. To remove the radical, square both sides of the equation: (dy/dx)^2 = x + d^2y/dx^2.<br>2. Rearrange the equation into a polynomial form: d^2y/dx^2 - (dy/dx)^2 + x = 0.<br>3. Identify the highest order derivative, which is d^2y/dx^2.<br>4. The order is 2.<br>5. The power of the highest order derivative (d^2y/dx^2) is 1.<br>6. Therefore, the degree is 1.
Final Answer: Order = 2, Degree = 1
Problem 255
Medium 1 Mark
Find the order and degree of the differential equation: (d^2y/dx^2)^3 + (dy/dx)^2 + y = 0
Show Solution
1. Identify the highest order derivative present in the equation. In this case, it is d^2y/dx^2.<br>2. The order is the order of this highest derivative, which is 2.<br>3. The equation is a polynomial in its derivatives. The power of the highest order derivative (d^2y/dx^2) is 3.<br>4. Therefore, the degree is 3.
Final Answer: Order = 2, Degree = 3
Problem 255
Easy 1 Mark
Find the order and degree of the differential equation: x(d^2y/dx^2) + y(dy/dx)^2 = 0
Show Solution
1. The highest order derivative is d^2y/dx^2. 2. The order is 2. 3. The equation is a polynomial in its derivatives. The power of the highest order derivative (d^2y/dx^2) is 1. 4. So, the degree is 1.
Final Answer: Order = 2, Degree = 1
Problem 255
Easy 1 Mark
State the order and degree of the differential equation: (d^3y/dx^3)^2 - 4(d^2y/dx^2) + 7y = sin(x)
Show Solution
1. The highest order derivative is d^3y/dx^3. 2. The order is 3. 3. The equation is a polynomial in its derivatives. The power of the highest order derivative (d^3y/dx^3) is 2. 4. So, the degree is 2.
Final Answer: Order = 3, Degree = 2
Problem 255
Easy 1 Mark
What are the order and degree of the differential equation: d^2y/dx^2 + sin(dy/dx) = 0?
Show Solution
1. Identify the highest order derivative: d^2y/dx^2. 2. The order is 2. 3. Check if the equation is a polynomial in its derivatives. The term sin(dy/dx) is a transcendental function of a derivative (dy/dx). 4. Because of sin(dy/dx), the equation is not a polynomial in its derivatives. 5. Therefore, the degree is not defined.
Final Answer: Order = 2, Degree = Not Defined
Problem 255
Easy 1 Mark
Find the order and degree of the differential equation: d^2y/dx^2 + (dy/dx)^(1/2) = 0
Show Solution
1. To find the degree, the differential equation must be a polynomial in its derivatives. Clear the fractional power by rearranging and squaring. 2. Rearrange: d^2y/dx^2 = -(dy/dx)^(1/2) 3. Square both sides: (d^2y/dx^2)^2 = (-(dy/dx)^(1/2))^2 => (d^2y/dx^2)^2 = dy/dx 4. Now, the equation is a polynomial in its derivatives: (d^2y/dx^2)^2 - dy/dx = 0. 5. The highest order derivative is d^2y/dx^2. 6. The order is 2. 7. The power of the highest order derivative (d^2y/dx^2) is 2. 8. So, the degree is 2.
Final Answer: Order = 2, Degree = 2
Problem 255
Easy 1 Mark
Determine the order and degree of the differential equation: dy/dx + cos(x) = 0
Show Solution
1. The highest order derivative is dy/dx. 2. The order is 1. 3. The power of dy/dx is 1. 4. The equation is a polynomial in its derivatives. So, the degree is 1.
Final Answer: Order = 1, Degree = 1

🎯IIT-JEE Main Problems (18)

Problem 255
Medium 4 Marks
If the order of the differential equation d^3y/dx^3 + (dy/dx)^4 - xy = 0 is 'p' and its degree is 'q', then the value of p x q is:
Show Solution
1. Identify the highest order derivative: d^3y/dx^3. Its order (p) is 3. 2. The equation is a polynomial in its derivatives. Identify the power of the highest order derivative: The power of (d^3y/dx^3) is 1. Its degree (q) is 1. 3. Calculate the product: p x q = 3 x 1 = 3.
Final Answer: 3
Problem 255
Hard 4 Marks
If y = 1 + (dy/dx) + (1/2!)(dy/dx)<sup>2</sup> + (1/3!)(dy/dx)<sup>3</sup> + ..., then the sum of the order and degree of the differential equation is:
Show Solution
1. Recognize the infinite series on the right-hand side. It is the Maclaurin series expansion for e<sup>x</sup>, where x is (dy/dx).<br>2. So, the given equation can be written as: y = e<sup>(dy/dx)</sup>.<br>3. To find the degree, we need to express this in a polynomial form of derivatives. Take the natural logarithm on both sides:<br> ln(y) = dy/dx<br>4. Now, identify the highest order derivative. It is dy/dx.<br>5. Therefore, the <b>order</b> of the differential equation is 1.<br>6. The highest order derivative dy/dx appears with a power of 1.<br>7. Therefore, the <b>degree</b> of the differential equation is 1.<br>8. The sum of the order and degree is 1 + 1 = 2.
Final Answer: 2
Problem 255
Hard 4 Marks
What is the degree of the differential equation ((d<sup>2</sup>y/dx<sup>2</sup>) + (dy/dx))<sup>2</sup> = (x + d<sup>3</sup>y/dx<sup>3</sup>)<sup>3/2</sup>?
Show Solution
1. Identify the highest order derivative in the equation. It is d<sup>3</sup>y/dx<sup>3</sup>. The order is 3.<br>2. To find the degree, eliminate the fractional power by squaring both sides of the equation:<br> (((d<sup>2</sup>y/dx<sup>2</sup>) + (dy/dx))<sup>2</sup>)<sup>2</sup> = ((x + d<sup>3</sup>y/dx<sup>3</sup>)<sup>3/2</sup>)<sup>2</sup><br>3. This simplifies to:<br> ((d<sup>2</sup>y/dx<sup>2</sup>) + (dy/dx))<sup>4</sup> = (x + d<sup>3</sup>y/dx<sup>3</sup>)<sup>3</sup><br>4. The equation is now a polynomial in its derivatives. The highest order derivative is d<sup>3</sup>y/dx<sup>3</sup>, and its power is 3.<br>5. Therefore, the <b>degree</b> of the differential equation is 3.
Final Answer: 3
Problem 255
Hard 4 Marks
The order and degree of the differential equation tan(d<sup>2</sup>y/dx<sup>2</sup>) + d<sup>3</sup>y/dx<sup>3</sup> = x<sup>2</sup>(dy/dx) are respectively:
Show Solution
1. Identify all derivatives present: dy/dx, d<sup>2</sup>y/dx<sup>2</sup>, d<sup>3</sup>y/dx<sup>3</sup>.<br>2. The highest order derivative is d<sup>3</sup>y/dx<sup>3</sup>.<br>3. Therefore, the <b>order</b> of the differential equation is 3.<br>4. To find the degree, check if the highest order derivative appears inside any transcendental function. In this equation, d<sup>2</sup>y/dx<sup>2</sup> is inside tan(...), but d<sup>3</sup>y/dx<sup>3</sup> (the highest order derivative) is not.<br>5. The term d<sup>3</sup>y/dx<sup>3</sup> itself is raised to the power of 1 (implicitly).<br>6. Since the highest order derivative is not an argument of a transcendental function and appears as a polynomial term, the equation can be considered a polynomial in d<sup>3</sup>y/dx<sup>3</sup>. The power of d<sup>3</sup>y/dx<sup>3</sup> is 1.<br>7. Therefore, the <b>degree</b> of the differential equation is 1.
Final Answer: Order = 3, Degree = 1
Problem 255
Hard 4 Marks
Find the degree of the differential equation: (d<sup>2</sup>y/dx<sup>2</sup>)<sup>2</sup> = (1 + (dy/dx)<sup>2</sup>)<sup>3/2</sup>.
Show Solution
1. Identify the highest order derivative. Here, it is d<sup>2</sup>y/dx<sup>2</sup>, so the order is 2.<br>2. To find the degree, eliminate the fractional power. Square both sides of the equation:<br> ((d<sup>2</sup>y/dx<sup>2</sup>)<sup>2</sup>)<sup>2</sup> = ((1 + (dy/dx)<sup>2</sup>)<sup>3/2</sup>)<sup>2</sup><br>3. This simplifies to:<br> (d<sup>2</sup>y/dx<sup>2</sup>)<sup>4</sup> = (1 + (dy/dx)<sup>2</sup>)<sup>3</sup><br>4. Now, the equation is a polynomial in its derivatives. The highest power of the highest order derivative (d<sup>2</sup>y/dx<sup>2</sup>) is 4.<br>5. Therefore, the <b>degree</b> of the differential equation is 4.
Final Answer: 4
Problem 255
Hard 4 Marks
Consider the differential equation: e<sup>(d<sup>3</sup>y/dx<sup>3</sup>)</sup> + x(d<sup>2</sup>y/dx<sup>2</sup>) = y(dy/dx). What is the order and degree of this equation?
Show Solution
1. Identify the highest order derivative in the equation. Here, it is d<sup>3</sup>y/dx<sup>3</sup>.<br>2. Therefore, the <b>order</b> of the differential equation is 3.<br>3. To find the degree, we check if the differential equation can be expressed as a polynomial in its derivatives.<br>4. The highest order derivative, d<sup>3</sup>y/dx<sup>3</sup>, appears inside an exponential function (e<sup>(...)</sup>).<br>5. An exponential function cannot be expressed as a finite polynomial in its argument.<br>6. Since the equation cannot be written as a polynomial in its derivatives, its <b>degree</b> is undefined.
Final Answer: Order = 3, Degree = Undefined
Problem 255
Hard 4 Marks
Determine the sum of the order and degree of the differential equation: (d<sup>3</sup>y/dx<sup>3</sup>)<sup>2/3</sup> + (d<sup>2</sup>y/dx<sup>2</sup>)<sup>2</sup> = (dy/dx)<sup>5</sup>.
Show Solution
1. Identify the highest order derivative in the equation. Here, it is d<sup>3</sup>y/dx<sup>3</sup>.<br>2. Therefore, the <b>order</b> of the differential equation is 3.<br>3. To find the degree, the equation must be free from radicals and fractional powers of derivatives. Rewrite the equation as: (d<sup>3</sup>y/dx<sup>3</sup>)<sup>2/3</sup> = (dy/dx)<sup>5</sup> - (d<sup>2</sup>y/dx<sup>2</sup>)<sup>2</sup>.<br>4. Cube both sides to eliminate the fractional power: ((d<sup>3</sup>y/dx<sup>3</sup>)<sup>2/3</sup>)<sup>3</sup> = ((dy/dx)<sup>5</sup> - (d<sup>2</sup>y/dx<sup>2</sup>)<sup>2</sup>)<sup>3</sup>.<br>5. This simplifies to: (d<sup>3</sup>y/dx<sup>3</sup>)<sup>2</sup> = ((dy/dx)<sup>5</sup> - (d<sup>2</sup>y/dx<sup>2</sup>)<sup>2</sup>)<sup>3</sup>.<br>6. Now, the highest power of the highest order derivative (d<sup>3</sup>y/dx<sup>3</sup>) is 2.<br>7. Therefore, the <b>degree</b> of the differential equation is 2.<br>8. The sum of the order and degree is 3 + 2 = 5.
Final Answer: 5
Problem 255
Medium 4 Marks
Consider the differential equation sqrt(1+(dy/dx)^2) = k (d^2y/dx^2), where k is a constant. The sum of the order and degree of this differential equation is:
Show Solution
1. Remove the fractional exponent (square root) by squaring both sides: 1+(dy/dx)^2 = k^2 (d^2y/dx^2)^2. 2. Identify the highest order derivative: d^2y/dx^2. Its order is 2. 3. Identify the power of the highest order derivative after making the equation a polynomial in derivatives: The power of (d^2y/dx^2) is 2. Its degree is 2. 4. Calculate the sum: 2 + 2 = 4.
Final Answer: 4
Problem 255
Medium 4 Marks
The degree of the differential equation d^2y/dx^2 + (dy/dx)^3 + y = x^2 is:
Show Solution
1. Identify the highest order derivative: d^2y/dx^2. Its order is 2. 2. The equation is a polynomial in its derivatives. Identify the power of the highest order derivative: The power of (d^2y/dx^2) is 1. Its degree is 1.
Final Answer: 1
Problem 255
Easy 4 Marks
Find the order and degree of the differential equation: (d³y/dx³)² + (d²y/dx²)³ + (dy/dx) + y = 0
Show Solution
1. Identify the highest order derivative present in the equation. Here, it is d³y/dx³. 2. The order of the differential equation is the order of this highest derivative, which is 3. 3. Check if the differential equation is a polynomial in its derivatives. Yes, it is. 4. The degree of the differential equation is the power of the highest order derivative when the equation is expressed as a polynomial in derivatives. The power of d³y/dx³ is 2.
Final Answer: Order = 3, Degree = 2
Problem 255
Medium 4 Marks
Let the order of the differential equation (d^2y/dx^2)^2 + (dy/dx)^3 = sin x be 'a' and its degree be 'b'. Then a+b is equal to:
Show Solution
1. Identify the highest order derivative: d^2y/dx^2. Its order (a) is 2. 2. The equation is a polynomial in its derivatives. Identify the power of the highest order derivative: The power of (d^2y/dx^2) is 2. Its degree (b) is 2. 3. Calculate the sum: a + b = 2 + 2 = 4.
Final Answer: 4
Problem 255
Medium 4 Marks
The sum of the order and degree of the differential equation d^2y/dx^2 = (y + dy/dx)^(1/2) is:
Show Solution
1. Remove the fractional exponent by squaring both sides: (d^2y/dx^2)^2 = y + dy/dx. 2. Identify the highest order derivative: d^2y/dx^2. Its order is 2. 3. Identify the power of the highest order derivative after making the equation a polynomial in derivatives: The power of (d^2y/dx^2) is 2. Its degree is 2. 4. Calculate the sum: 2 + 2 = 4.
Final Answer: 4
Problem 255
Medium 4 Marks
The order and degree of the differential equation (1+(dy/dx)^2)^(3/2) = d^2y/dx^2 are respectively:
Show Solution
1. Remove the fractional exponent by squaring both sides: (1+(dy/dx)^2)^3 = (d^2y/dx^2)^2. 2. Identify the highest order derivative: d^2y/dx^2. Its order is 2. 3. Identify the power of the highest order derivative after making the equation a polynomial in derivatives: The power of (d^2y/dx^2) is 2. Its degree is 2.
Final Answer: Order = 2, Degree = 2
Problem 255
Easy 4 Marks
Determine the order and degree of the differential equation: √(dy/dx) = (d²y/dx²) + 5
Show Solution
1. To define the degree, remove the fractional power. Square both sides: [√(dy/dx)]² = [(d²y/dx²) + 5]². 2. This simplifies to dy/dx = (d²y/dx²)² + 10(d²y/dx²) + 25. 3. The highest order derivative present is d²y/dx². 4. The order of the differential equation is 2. 5. The equation is now a polynomial in its derivatives. The power of the highest order derivative (d²y/dx²) is 2.
Final Answer: Order = 2, Degree = 2
Problem 255
Easy 4 Marks
What is the sum of the order and degree of the differential equation: y = x(dy/dx) + (dy/dx)³?
Show Solution
1. Identify the highest order derivative: dy/dx. 2. The order is 1. 3. The equation is a polynomial in its derivatives. 4. The highest power of the highest order derivative (dy/dx) is 3. 5. The degree is 3. 6. Sum = Order + Degree = 1 + 3 = 4.
Final Answer: Order = 1, Degree = 3, Sum = 4
Problem 255
Easy 4 Marks
Find the order and degree of the differential equation: (dy/dx)⁴ + 3y(d²y/dx²) = 0
Show Solution
1. Identify the highest order derivative: d²y/dx². 2. The order is 2. 3. The equation is a polynomial in its derivatives. 4. The power of the highest order derivative (d²y/dx²) is 1.
Final Answer: Order = 2, Degree = 1
Problem 255
Easy 4 Marks
What are the order and degree of the differential equation: d²y/dx² + cos(dy/dx) = 0?
Show Solution
1. Identify the highest order derivative: d²y/dx². 2. The order is 2. 3. Check if the equation is a polynomial in its derivatives. Due to the term cos(dy/dx), the equation is not a polynomial in its derivatives. 4. Since it is not a polynomial in its derivatives, the degree is not defined.
Final Answer: Order = 2, Degree = Not Defined
Problem 255
Easy 4 Marks
Determine the order and degree of the differential equation: (d²y/dx²)^(3/2) = 1 + (dy/dx)²
Show Solution
1. To define the degree, first remove any fractional powers by suitable algebraic manipulation. Square both sides of the equation: [(d²y/dx²)^(3/2)]² = (1 + (dy/dx)²)². 2. This simplifies to (d²y/dx²)³ = (1 + (dy/dx)²)². 3. The highest order derivative present is d²y/dx². 4. The order of the differential equation is 2. 5. The equation is now a polynomial in its derivatives. The power of the highest order derivative (d²y/dx²) is 3.
Final Answer: Order = 2, Degree = 3

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📐Important Formulas (3)

Order of a Differential Equation
ext{Order} = ext{Highest order derivative present}
Text: Order = Highest order derivative present in the differential equation.
The <b>order</b> of a differential equation is defined as the order of the highest order derivative appearing in the equation. It's a fundamental property classifying ODEs.<br><ul><li>For example, in $ frac{d^2y}{dx^2} + x frac{dy}{dx} + y = 0 $, the highest order derivative is $ frac{d^2y}{dx^2} $, which is a second-order derivative. Hence, the order of this differential equation is 2.</li><li>The order is always a <strong>positive integer</strong>.</li><li>This concept is crucial for both <strong>CBSE 12th Board</strong> and <strong>JEE Main/Advanced</strong>.</li></ul>
Variables: Always used to classify a differential equation based on the highest differentiation performed on the dependent variable.
Degree of a Differential Equation (when defined)
ext{Degree} = ext{Power of the highest order derivative}
Text: Degree = The power of the highest order derivative when the differential equation is a polynomial in its derivatives.
The <b>degree</b> of a differential equation is defined as the power of the highest order derivative, <span style='color: #FF0000;'>provided that the differential equation can be expressed as a polynomial in its derivatives</span>. <br><ul><li>If the equation is polynomial in derivatives, identify the highest order derivative. Its exponent (power) is the degree.</li><li>For example, in $ left( frac{d^2y}{dx^2} ight)^3 + x left( frac{dy}{dx} ight)^2 + y = 0 $, the highest order derivative is $ frac{d^2y}{dx^2} $ and its power is 3. So, the degree is 3.</li><li>The degree is always a <strong>positive integer</strong>.</li><li>This is a key concept for <strong>CBSE 12th Board</strong> and <strong>JEE Main/Advanced</strong>.</li></ul>
Variables: Used to classify a differential equation, but only after confirming it's a polynomial in its derivatives. This is the second step after finding the order.
Condition for Degree to be Defined
ext{Degree is defined if ODE is a polynomial in its derivatives}
Text: Degree is defined only if the differential equation is expressible as a polynomial in all its derivatives.
The degree of a differential equation is <b>not always defined</b>. It is defined <span style='color: #0000FF;'>only if the differential equation can be written as a polynomial in all its derivatives</span>. <br><ul><li>This means there should be <strong>no fractional powers</strong> of derivatives ($ (y')^{1/2}, (y'')^{3/4} $).</li><li>There should be <strong>no transcendental functions</strong> of derivatives ($ sin(y'), e^{y''}, log(y') $).</li><li>If these conditions are not met, the degree is said to be <strong>'not defined'</strong>. This distinction is particularly important for <strong>JEE Main/Advanced</strong> where such cases are frequently tested.</li></ul>
Variables: This condition must be checked before attempting to state the degree of any differential equation. If the condition is not met, the degree is 'not defined'.

📚References & Further Reading (10)

Book
Ordinary and Partial Differential Equations
By: M. D. Raisinghania
https://www.schan.in/product/ordinary-and-partial-differential-equations-18th-edition/
An extensively used textbook in India for university-level mathematics courses. The initial chapter, 'Introduction to Differential Equations', thoroughly covers the basic definitions, including the order and degree of ODEs, with numerous solved examples.
Note: Highly relevant for Indian students, covers the topic in detail with an exam-oriented approach. Essential for understanding the basic classification of ODEs.
Book
By:
Website
Introduction to differential equations
By: Khan Academy
https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/modal/v/introduction-to-differential-equations
This introductory video and accompanying notes explain what differential equations are and how to identify their order and degree using graphical and algebraic examples.
Note: Visually engaging and conceptually clear. Great for initial understanding and solidifying definitions through practical examples. Good for CBSE and JEE basic concepts.
Website
By:
PDF
Differential Equations (NPTEL Lecture 1)
By: Prof. S. K. Gupta
https://nptel.ac.in/content/storage2/courses/111105072/pdf/lec_1.pdf
The inaugural lecture notes from the NPTEL course on Differential Equations. It meticulously defines ordinary and partial differential equations, elaborates on their order, degree, and linearity with detailed examples.
Note: An authoritative Indian academic resource, highly recommended for a thorough understanding suitable for JEE Advanced. Covers all necessary definitions.
PDF
By:
Article
Differential Equation Order
By: Eric W. Weisstein
https://mathworld.wolfram.com/DifferentialEquationOrder.html
Wolfram MathWorld provides a concise definition and explanation of the 'order' of a differential equation, linking it to the highest derivative present. While separate articles define 'degree', this focuses purely on order.
Note: A precise, technical definition suitable for reinforcing understanding of the exact meaning of 'order' in ODEs, useful for advanced conceptual clarity.
Article
By:
Research_Paper
Historical Development of the Theory of Differential Equations
By: A. D. Myshkis
https://encyclopediaofmath.org/wiki/Historical_development_of_the_theory_of_differential_equations
This article, though not a traditional research paper, provides a comprehensive historical account of differential equations. It traces the evolution of concepts, including the early classification and understanding of ODEs based on properties like order, highlighting how these fundamental definitions emerged.
Note: Provides historical context for the concepts of order and degree, illustrating their fundamental nature in the development of ODE theory. Good for a deeper, conceptual appreciation.
Research_Paper
By:

⚠️Common Mistakes to Avoid (57)

Minor Other

❌ Incorrect Degree Determination for Non-Polynomial Differential Equations

Students frequently make errors in determining the degree of an ordinary differential equation (ODE) when the equation is not presented in a polynomial form with respect to its derivatives. This often occurs when derivatives have fractional or negative powers, or are arguments of transcendental functions (like sine, cosine, logarithm, exponential). A common mistake is to directly state a fractional power as the degree or to attempt to define degree when it's genuinely undefined.
💭 Why This Happens:
This mistake stems from a lack of thorough understanding of the precise definition of 'degree'. The degree is only defined if the ODE can be expressed as a polynomial in its derivatives. Students often rush to identify the degree without first simplifying the equation to eliminate fractional/negative powers of derivatives or recognizing when such simplification is impossible.
✅ Correct Approach:
To correctly determine the degree, follow these steps:
  1. First, identify the order of the ODE, which is the order of the highest derivative present.
  2. Next, clear all fractional and negative powers of derivatives by suitable algebraic manipulations (e.g., squaring both sides, raising to a certain power). The goal is to express the ODE as a polynomial in its derivatives.
  3. Once the equation is in polynomial form with respect to its derivatives, the power of the highest order derivative is its degree.
  4. If the ODE cannot be expressed as a polynomial in its derivatives (e.g., if a derivative is inside a trigonometric, exponential, or logarithmic function), then the degree is undefined.
📝 Examples:
❌ Wrong:
Consider the ODE: (d²y/dx²) = √[ (dy/dx) + x ]
A common wrong approach: Identifying order as 2 and attempting to state the degree as 1 directly without rationalization, or even incorrectly stating 1/2 as the degree of (dy/dx).
✅ Correct:
For the ODE: (d²y/dx²) = √[ (dy/dx) + x ]
  1. The highest derivative is d²y/dx², so the order is 2.
  2. To make it a polynomial, square both sides: (d²y/dx²)² = (dy/dx) + x.
  3. Now, the equation is a polynomial in its derivatives. The highest order derivative is d²y/dx², and its power is 2. Therefore, the degree is 2.

Another example: e^(dy/dx) + d²y/dx² = 0
Here, dy/dx is an argument of the exponential function. This ODE cannot be expressed as a polynomial in its derivatives. Therefore, its degree is undefined.
💡 Prevention Tips:
  • Always rationalize fractional/negative powers of derivatives before determining the degree.
  • Be vigilant for derivatives inside transcendental functions (e.g., sin(y'), log(y''), e^(y''')); if present, the degree is undefined.
  • Practice algebraic manipulation to convert ODEs into polynomial form.
  • Remember the fundamental definition: degree is the power of the highest order derivative after the equation is made free from radicals and fractions as far as derivatives are concerned.
JEE_Advanced
Minor Conceptual

❌ Incorrectly Determining Degree Due to Misunderstanding 'Polynomial in Derivatives' Condition

Students often make mistakes in determining the degree of an ordinary differential equation (ODE) by directly picking the highest power of the highest order derivative without first ensuring that the equation is a polynomial in its derivatives. This leads to errors when fractional powers of derivatives are present or when derivatives appear inside non-algebraic functions like trigonometric, exponential, or logarithmic functions.
💭 Why This Happens:
This mistake stems from a fundamental misunderstanding of the precise definition of the 'degree' of a differential equation. Many students remember 'highest power of the highest order derivative' but overlook the crucial prerequisite that the equation must be expressible as a polynomial in all its derivatives. Consequently, they fail to clear fractional powers or recognize instances where the degree simply isn't defined.
✅ Correct Approach:
To correctly determine the order and degree of an ODE for JEE Main, follow these steps:
  • Order: Identify the highest order derivative present in the equation. This determines the order.
  • Degree (if defined):
    1. First, ensure the differential equation is free from radicals (fractional powers) involving derivatives. If not, manipulate the equation (e.g., by squaring both sides) to clear them.
    2. Verify that the equation can be expressed as a polynomial in all its derivatives (y', y'', y'''...). This means no derivatives should be inside functions like sin(), cos(), e^(), log(), etc.
    3. If the equation is a polynomial in its derivatives, the degree is the highest power of the highest order derivative.
    4. Important: If a derivative is inside a non-algebraic function (e.g., sin(y''), e^(y'), log(y''')), then the degree is not defined.
📝 Examples:
❌ Wrong:
Consider the equation: (y''')^(2/3) + y' = 0
A common mistake is to state that the order is 3 and the degree is 2/3. This is incorrect because the equation is not a polynomial in its derivatives due to the fractional power.
✅ Correct:
For the equation: (y''')^(2/3) + y' = 0
  1. Order: The highest order derivative is y''', so the order is 3.
  2. Degree: To find the degree, first isolate the fractional power and clear it:
    (y''')^(2/3) = -y'
    Now, cube both sides to remove the fractional power:
    ((y''')^(2/3))^3 = (-y')^3
    (y''')^2 = -(y')^3
    The equation is now (y''')^2 + (y')^3 = 0. This is a polynomial in its derivatives.
    The highest order derivative is y''', and its highest power is 2. Therefore, the degree is 2.

Another important case: For sin(y'') + y = 0, the order is 2, but the degree is not defined because y'' is inside a sine function.
💡 Prevention Tips:
  • Rigorous Definition: Always recall the full definition of degree: 'The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in its derivatives.'
  • Check for Radicals: Systematically remove all fractional powers involving derivatives by algebraic manipulation (e.g., squaring, cubing) before determining the degree.
  • Beware of Transcendentals: If any derivative (y', y'', etc.) is an argument of a transcendental function (sin, cos, tan, log, e^), then the degree is not defined. This is a common trap in JEE Main.
JEE_Main
Minor Calculation

❌ Incorrect Degree Calculation with Fractional Powers/Radicals

Students often calculate the degree of an ODE directly from its given form, failing to ensure it's a polynomial in derivatives. This error frequently occurs when fractional powers or radicals involving derivative terms are present.
💭 Why This Happens:
This error stems from an incomplete understanding of the definition of degree. The degree is defined as the power of the highest order derivative after the differential equation has been made free from radicals and fractional powers of the derivatives. Students often skip this critical simplification step, directly picking the exponent from the unsimplified form.
✅ Correct Approach:
Always ensure the differential equation is a polynomial in its derivatives. If there are fractional powers or radicals involving derivative terms, eliminate them by raising both sides to appropriate integer powers (e.g., squaring, cubing, etc.). This might require isolating the term with the fractional power first. Only after this simplification can you correctly identify the power of the highest order derivative as the degree.
📝 Examples:
❌ Wrong:

Consider the ODE: dy/dx + (d²y/dx²)^(1/3) = 0

Wrong approach:

  • Highest order derivative is d²y/dx².
  • Its power is 1/3.
  • Incorrectly concludes: Degree = 1/3.
✅ Correct:

Consider the ODE: dy/dx + (d²y/dx²)^(1/3) = 0

Correct approach:

  1. Isolate the term with the fractional power: (d²y/dx²)^(1/3) = -dy/dx
  2. To clear the fractional power (1/3), cube both sides of the equation: ((d²y/dx²)^(1/3))³ = (-dy/dx)³
  3. This simplifies to: d²y/dx² = -(dy/dx)³
  4. Now, the equation is a polynomial in its derivatives.
  5. The highest order derivative is d²y/dx², and its power in this polynomial form is 1.
  6. Therefore, Order = 2, Degree = 1.

JEE Note: For JEE Main, correctly identifying the degree after clearing radicals/fractions is a common check. Always simplify first!

💡 Prevention Tips:
  • Know the Definition: Revisit the precise definition of 'degree' for a differential equation, emphasizing the 'polynomial in derivatives' condition.
  • Scan for Fractions/Radicals: Before determining the degree, always scrutinize the equation for any derivative terms under radicals or with fractional exponents.
  • Systematic Simplification: If detected, make it a habit to systematically eliminate these fractional powers/radicals by appropriate algebraic manipulation (e.g., isolating and raising to an integer power).
  • Practice: Work through multiple problems involving ODEs with radicals and fractional powers to reinforce this crucial step.
JEE_Main
Minor Formula

❌ Ignoring Conditions for Degree Definition

Students frequently attempt to determine the degree of a differential equation without first verifying if it can be expressed as a polynomial in its derivatives. This oversight often leads to an incorrect or undefined degree being stated.
💭 Why This Happens:
This mistake stems from a fundamental misunderstanding of the prerequisites for a differential equation to possess a defined degree. Students sometimes confuse 'order' (which is always defined) with 'degree' (which has specific defining conditions). They may overlook the critical rule that the differential equation must be a polynomial in all its derivatives, free from radicals, fractional powers, or transcendental functions involving derivatives.
✅ Correct Approach:
To correctly determine the degree, always follow these steps:
  1. First, identify the order: the highest order derivative present in the equation.
  2. Next, check if the differential equation can be expressed as a polynomial in its derivatives. This means clearing any radicals or fractional powers of derivatives by raising both sides to an appropriate power. If derivatives appear within transcendental functions (e.g., sin(y'), e^(y''), log(y')), or if fractional powers cannot be eliminated, then the equation is not a polynomial in its derivatives, and the degree is undefined.
  3. If it is a polynomial, the degree is the power of the highest order derivative after clearing all radicals and fractions.
📝 Examples:
❌ Wrong:
For the equation sin(y') + (y'')^3 + y = 0, a student might incorrectly state the degree as 3, considering the power of y''.
✅ Correct:
Consider the equation sin(y') + (y'')^3 + y = 0.
The order is 2 (due to y'').
However, the term sin(y') involves a derivative inside a transcendental function. This means the equation cannot be expressed as a polynomial in its derivatives. Therefore, the degree is undefined.
💡 Prevention Tips:
  • Verify Polynomial Form: Always check if the differential equation is a polynomial in its derivatives (i.e., free from derivatives inside trigonometric, exponential, or logarithmic functions, and free from fractional/negative powers of derivatives that cannot be rationalized).
  • Clear Radicals/Fractions: If radicals or fractional powers are present, raise both sides of the equation to an appropriate power to eliminate them *before* determining the degree. This step is crucial for JEE Main problems.
  • Distinguish Order vs. Degree: Remember that order is almost always defined, but degree has strict conditions. Don't assume a degree always exists.
  • Practice Critical Cases: Focus on problems where the degree is undefined to build a strong conceptual understanding.
JEE_Main
Minor Unit Conversion

❌ Neglecting 'Polynomial Form' for Degree

Students frequently determine the degree of an Ordinary Differential Equation (ODE) incorrectly by not first 'converting' it to a form that is a polynomial in all its derivatives. This overlooks a fundamental requirement for the degree's definition and calculation.
💭 Why This Happens:
Errors arise when students prematurely identify the highest derivative's power without performing necessary algebraic manipulations. They fail to eliminate non-polynomial terms like radicals (e.g., √(d²y/dx²)) or transcendental functions of derivatives (e.g., sin(dy/dx)). The crucial 'conversion' to the polynomial standard form, essential for defining degree, is often missed or done incorrectly.
✅ Correct Approach:
To accurately find the degree, algebraically manipulate the ODE to remove all radicals involving derivatives and express it as a polynomial in all derivatives. Only after this 'conversion' should the highest power of the highest order derivative be identified. If the equation cannot be made polynomial in its derivatives (e.g., due to sin(dy/dx) or log(d²y/dx²) terms), then the degree is undefined. This rule applies consistently for both CBSE board exams and JEE Main.
📝 Examples:
❌ Wrong:
Consider the ODE: d³y/dx³ + √(d²y/dx²) + dy/dx = 0.
A common error is to state the highest order as 3 and incorrectly claim the degree is 1 (from d³y/dx³) or 1/2 (from √(d²y/dx²)), thereby skipping the necessary 'conversion' process to a polynomial form.
✅ Correct:
Given the ODE: d³y/dx³ + √(d²y/dx²) + dy/dx = 0.
  1. Isolate the radical term: d³y/dx³ + dy/dx = -√(d²y/dx²).
  2. Square both sides to eliminate the radical: (d³y/dx³ + dy/dx)² = d²y/dx².
Now, the equation is a polynomial in its derivatives. The highest order derivative is d³y/dx³ (order = 3), and its highest power in this polynomial form is 2. Thus, the order is 3 and the degree is 2.
💡 Prevention Tips:
  • Always check Polynomial Form: Before determining the degree, ensure the ODE is algebraically 'converted' into a polynomial in its derivatives.
  • Master Algebraic Transformations: Practice skills like squaring or cubing to effectively eliminate radicals involving derivatives.
  • Recognize Undefined Degree: Be aware that if an ODE contains terms like sin(dy/dx) or log(d²y/dx²), its degree is undefined, as it cannot be made into a polynomial in derivatives.
JEE_Main
Minor Sign Error

❌ Sign Errors in Algebraic Manipulation for Degree Calculation

Students often make sign errors during the algebraic manipulation required to transform a differential equation into a polynomial in its derivatives, especially when dealing with negative signs outside or inside fractional powers and radicals. This error can lead to an incorrect final form of the equation, making it difficult to correctly identify the degree.
💭 Why This Happens:
This mistake typically arises from carelessness or a lack of attention to detail during algebraic steps such as squaring, cubing, or raising both sides of an equation to a specific power. Students might incorrectly handle the negative sign when it's part of a term being raised to an even or odd power, or when transposing terms across the equality. For instance, confusing `-(A)^2` with `(-A)^2` or mishandling signs during operations like `(-X)^(1/3)` vs `-(X)^(1/3)` after cubing.
✅ Correct Approach:
Always ensure the differential equation is a polynomial in its derivatives before determining its degree. Pay meticulous attention to algebraic operations, particularly when eliminating fractional powers or radicals by raising both sides to a power. Remember that `(-A)^2 = A^2` (for even powers) and `(-A)^3 = -A^3` (for odd powers). Double-check each step of algebraic manipulation, especially when signs are involved, to maintain the correct form of the equation.
📝 Examples:
❌ Wrong:
Consider the ODE:
(d^2y/dx^2) = - (dy/dx)^(1/3)
To find the degree, we cube both sides to eliminate the fractional power:
(d^2y/dx^2)^3 = [ - (dy/dx)^(1/3) ]^3
Common Sign Error: A student might incorrectly write the RHS as - (dy/dx) (if they mistakenly interpret `[ - (dy/dx)^(1/3) ]^3` as `-[ (dy/dx)^(1/3) ]^3`), or sometimes even as (dy/dx) if they drop the negative sign entirely before cubing.
This leads to the incorrect form: (d^2y/dx^2)^3 = - (dy/dx) (incorrect simplification of RHS) or (d^2y/dx^2)^3 = (dy/dx) (dropping the negative sign incorrectly).
✅ Correct:
For the same ODE:
(d^2y/dx^2) = - (dy/dx)^(1/3)
Cube both sides correctly:
(d^2y/dx^2)^3 = [ - (dy/dx)^(1/3) ]^3
Applying the rule `(-A)^3 = -A^3`:
(d^2y/dx^2)^3 = - (dy/dx)
Rearranging to the polynomial form:
(d^2y/dx^2)^3 + (dy/dx) = 0
Here, the highest order derivative is d^2y/dx^2 (order 2), and its highest power is 3. Thus, the degree is 3.

Note for JEE Main: While the final degree often remains the same even with a sign error in a coefficient, a sign error can lead to a misformed polynomial, causing confusion or doubt about the equation's polynomial nature or the correct identification of the highest power.
💡 Prevention Tips:
  • Be thorough: Always write down all intermediate algebraic steps, especially when clearing radicals or fractional powers.
  • Check signs: Pay extra attention to negative signs, ensuring they are correctly applied when terms are squared, cubed, or moved across the equality.
  • Recall basic algebra: Revisit rules for exponents with negative bases, e.g., `(-x)^n`.
  • Verify polynomial form: After all manipulations, ensure the equation truly represents a polynomial in terms of its derivatives.
JEE_Main
Minor Approximation

❌ Ignoring Fractional Powers/Radicals when Determining Degree

Students frequently make an error by directly identifying the degree of a differential equation without first eliminating fractional powers or radicals that involve derivatives. This 'approximated' understanding leads to an incorrect degree, or sometimes even an incorrect conclusion about the degree being undefined.
💭 Why This Happens:
This mistake stems from an incomplete understanding or a hasty application of the definition of the degree of a differential equation. Students often forget the critical prerequisite: the differential equation must be expressible as a polynomial in all its derivatives for its degree to be defined. Fractional powers or radicals prevent it from being a polynomial.
✅ Correct Approach:

Before determining the degree of a differential equation, it is crucial to ensure that it is free from fractional powers or radicals involving any derivatives.

To achieve this, manipulate the equation by isolating the term with the fractional exponent or radical and then raising both sides to an appropriate integer power to clear the fractional exponent. Only after the equation is a polynomial in its derivatives can you correctly identify the highest power of the highest order derivative as the degree.

JEE Tip: Always simplify the equation to its polynomial form in derivatives before stating the degree. For CBSE, this step is equally vital.

📝 Examples:
❌ Wrong:

Consider the differential equation: (d2y/dx2)3/2 = dy/dx + x

A common mistake is to directly state the degree as 3/2, or incorrectly conclude that the degree is undefined because of the fractional power without attempting simplification.

✅ Correct:

For the equation: (d2y/dx2)3/2 = dy/dx + x

1. Square both sides to eliminate the fractional power:

((d2y/dx2)3/2)2 = (dy/dx + x)2

(d2y/dx2)3 = (dy/dx + x)2

2. Now, the equation is a polynomial in its derivatives. The highest order derivative is d2y/dx2 (order 2), and its highest power is 3.

Therefore, the order is 2 and the degree is 3.

💡 Prevention Tips:
  • Always Rationalize: Before finding the degree, make sure all derivatives are raised to integer powers. Eliminate any radicals or fractional exponents by appropriate algebraic manipulation (e.g., squaring both sides, cubing both sides, etc.).
  • Polynomial in Derivatives: Remember that the degree is defined only if the differential equation is a polynomial in its derivatives. If terms like sin(dy/dx) or e^(d²y/dx²) are present, the degree is undefined, and no amount of simplification will change that.
  • Verify Highest Power: After rationalizing, identify the highest order derivative and then look for its highest power.
JEE_Main
Minor Other

❌ Incorrectly Assigning Degree to Non-Polynomial Differential Equations

A common minor mistake is attempting to find the 'degree' of a differential equation when it is not a polynomial in its derivatives. Students often overlook the crucial condition that for the degree to be defined, the differential equation must be expressible as a polynomial in all its derivatives (dy/dx, d²y/dx², etc.). If terms like sin(y'), e^(y''), log(y'), etc., are present, the degree is undefined.
💭 Why This Happens:
This mistake stems from an incomplete understanding of the precise definition of 'degree'. While the 'order' is always defined as the order of the highest derivative present, the 'degree' has an additional prerequisite: the equation must be free from fractional powers of derivatives and transcendental functions of derivatives. Students often apply the 'highest power' rule without first verifying this foundational condition.
✅ Correct Approach:
Always first determine the order of the differential equation. Then, check if the equation can be written as a polynomial in its derivatives. If it contains derivatives inside transcendental functions (like sin, cos, e^, log, etc.) or has fractional powers of derivatives that cannot be cleared, then the degree is undefined. Only if it is a polynomial in its derivatives can you find the power of the highest order derivative after making the equation free from radicals/fractional powers.
📝 Examples:
❌ Wrong:
Consider the equation: d²y/dx² + cos(dy/dx) = 0.
A student might incorrectly identify the order as 2 and then look for the power of d²y/dx², finding it to be 1, and thus declare the degree as 1.
✅ Correct:
For the equation: d²y/dx² + cos(dy/dx) = 0.
  • The highest order derivative is d²y/dx², so the Order = 2.
  • However, the term cos(dy/dx) means the equation is NOT a polynomial in its derivatives.
  • Therefore, the Degree is Undefined.
JEE Tip: Both Order and Degree are important for JEE Main. While order is straightforward, degree often involves this critical check.
💡 Prevention Tips:
  • Check for Transcendental Functions: Before determining degree, scan the equation for terms like sin(y'), e^(y''), log(y'), etc. If found, the degree is undefined.
  • Clear Radicals/Fractional Powers: If derivatives are under radicals or have fractional powers, square/cube/raise to an appropriate power to make them integers, then find the degree. This must be done carefully without introducing new derivatives.
  • Recall Definition: Reiterate that 'degree' is the power of the highest order derivative, *provided the equation is a polynomial in its derivatives*.
JEE_Main
Minor Other

❌ Incorrectly Assigning Degree to ODEs Not Polynomial in Derivatives

Students frequently make the error of stating a definite degree for an Ordinary Differential Equation (ODE) even when it cannot be expressed as a polynomial in its derivatives. This occurs when derivatives are embedded within transcendental functions.
💭 Why This Happens:
This mistake stems from a superficial understanding of the degree definition. Many students overlook the crucial condition that an ODE's degree is only defined if it can be written as a polynomial in its derivatives. They often rush, applying the 'highest power of the highest order derivative' rule without checking this prerequisite.
✅ Correct Approach:
To correctly determine the order and degree of an ODE:
  1. First, identify the order, which is the order of the highest derivative present in the equation.
  2. Next, critically examine if the ODE can be expressed as a polynomial in all its derivatives. This means no derivative term should be an argument of a transcendental function (e.g., sin(y'), e^(y''), log(y''')).
  3. If it can be written as a polynomial in derivatives (after clearing radicals/fractions involving derivatives), then the degree is the highest power of the highest order derivative.
  4. If it cannot be expressed as a polynomial in its derivatives, then the degree is not defined. This is a common outcome for many ODEs in JEE and CBSE.
📝 Examples:
❌ Wrong:
Consider the ODE: sin(d²y/dx²) + (dy/dx)³ = 0.
A common incorrect approach is to state the order as 2 and the degree as 1 (from sin(d²y/dx²)) or 3 (from (dy/dx)³).
✅ Correct:
For the ODE: sin(d²y/dx²) + (dy/dx)³ = 0.
  • Order: The highest order derivative is d²y/dx², so the order is 2.
  • Degree: The term sin(d²y/dx²) involves the second derivative inside a sine function. This prevents the entire equation from being expressed as a polynomial in its derivatives. Therefore, the degree is not defined.
💡 Prevention Tips:
  • Key Check: Always verify if the ODE is a polynomial in derivatives before attempting to find its degree.
  • Be particularly cautious when derivatives appear inside trigonometric (sin, cos, tan), exponential (e^), or logarithmic (log) functions.
  • Remember that 'degree not defined' is a valid and often correct answer, especially in more complex ODEs.
  • For CBSE Class 12, this specific condition is frequently tested to check a student's precise understanding of definitions.
CBSE_12th
Minor Sign Error

❌ <span style='color: #FF0000;'>Incorrect Handling of Negative Signs during Radical/Fractional Power Elimination for Degree Calculation</span>

Students sometimes make a subtle sign error when squaring or cubing both sides of an ordinary differential equation (ODE) to eliminate radicals or fractional powers. This often occurs when one side of the equation contains a negative sign or is a negative expression. While this specific error might not always alter the final numerical value of the order or degree, it indicates a fundamental algebraic oversight and can lead to incorrect intermediate steps, especially if the equation needs to be fully expanded for other purposes.
💭 Why This Happens:
  • Carelessness: Rushing through algebraic manipulations without careful attention to signs.
  • Confusion of Properties: Not clearly distinguishing between expressions like (-A)2 (which is A2) and -(A2) (which remains negative).
  • Lack of Focus: Failing to apply the exponent to the entire expression on a side of the equation, including any leading negative signs.
✅ Correct Approach:
Always ensure that the entire side of the equation, including any negative signs, is raised to the power required to clear the radical or fractional exponent. Remember the rules of exponents for negative bases: for an even exponent 'n', (-A)n = An; for an odd exponent 'n', (-A)n = -An. The ultimate goal for determining degree is to obtain a polynomial in derivatives with non-negative integer powers.
📝 Examples:
❌ Wrong:

Consider the ODE: (dy/dx)1/2 = -(x + y)

Incorrect Step: A student might square both sides and write:

dy/dx = -(x + y)2

Here, the negative sign is wrongly kept outside the squared term, implying -1 * (x+y)2 instead of (-1 * (x+y))2.

Order: 1, Degree: 1 (The mistake doesn't change the final order/degree, but the algebraic form is wrong).

✅ Correct:

Consider the ODE: (dy/dx)1/2 = -(x + y)

Correct Step: Square both sides, treating -(x+y) as a single term:

( (dy/dx)1/2 )2 = ( -(x + y) )2
dy/dx = (x + y)2
dy/dx = x2 + 2xy + y2

Order: 1, Degree: 1.

💡 Prevention Tips:
  • Bracket Carefully: When raising an entire side of an equation to a power, mentally or physically enclose that side in brackets to ensure the exponent applies to everything.
  • Know Exponent Rules: Revisit and reinforce understanding of how negative signs behave under even and odd exponents.
  • CBSE vs. JEE: This algebraic precision is crucial for both CBSE and JEE. While a minor sign error might not alter order/degree, it can lead to marks deduction in descriptive steps (CBSE) or propagate errors in more complex problems (JEE).
  • Verify Expansion: If the equation needs further expansion, a sign error here will definitely lead to an incorrect final form.
CBSE_12th
Minor Formula

❌ Ignoring Fractional Powers/Radicals for Degree Calculation

Students often directly state the degree of a differential equation without first ensuring it is expressed as a polynomial in its derivatives, especially when simple fractional powers or radicals are present on derivative terms. They might incorrectly take the power of the highest order derivative without clearing these non-integer exponents.
💭 Why This Happens:
This mistake primarily stems from a lack of careful application of the definition of 'degree'. Students might rush, overlook the crucial condition that the differential equation must be expressible as a polynomial in its derivatives, or forget basic algebraic manipulations to remove fractional powers (e.g., squaring both sides).
✅ Correct Approach:
Always make sure the differential equation is free from radicals and fractional powers of the derivatives before determining its degree. This usually involves isolating the terms with fractional powers/radicals and raising both sides of the equation to an appropriate power to eliminate them. Once the equation is a polynomial in its derivatives, identify the highest power of the highest order derivative.
📝 Examples:
❌ Wrong:
Consider the equation: (d²y/dx²)³ᐟ² + dy/dx = 0
A common mistake is to state the order is 2 and the degree is 3/2, directly from the highest derivative term.
✅ Correct:
For the equation: (d²y/dx²)³ᐟ² + dy/dx = 0
  1. First, isolate the term with the fractional power: (d²y/dx²)³ᐟ² = -dy/dx
  2. To remove the fractional power, square both sides: ((d²y/dx²)³ᐟ²)² = (-dy/dx)²
  3. This simplifies to: (d²y/dx²)³ = (dy/dx)²
Now, the equation is a polynomial in its derivatives.
The highest order derivative is d²y/dx², so the Order = 2.
The power of the highest order derivative (d²y/dx²) is 3, so the Degree = 3.
💡 Prevention Tips:
  • Remember the Golden Rule for Degree: The differential equation must be a polynomial in its derivatives before determining the degree.
  • Always Simplify: Before finding the degree, ensure there are no square roots, cube roots, or fractional powers over any derivative term.
  • Practice Algebraic Manipulation: Be adept at squaring/cubing both sides of an equation to remove radicals.
  • CBSE/JEE Alert: This is a very common trap in both Board exams and competitive exams. Always double-check this condition.
CBSE_12th
Minor Calculation

❌ Misinterpreting Degree when ODE is not a Polynomial in Derivatives

Students frequently make the mistake of assigning a degree to a differential equation even when it cannot be expressed as a polynomial in its derivatives. This fundamental misunderstanding of the definition of 'degree' leads to incorrect answers, especially in CBSE exams where such conceptual clarity is tested.
💭 Why This Happens:
This error stems from a lack of complete understanding of the definition of 'degree'. While order is always defined for any ODE, degree is only defined if the equation is a polynomial in its derivatives. Students often confuse this and attempt to find the power of the highest order derivative regardless of whether the equation satisfies the polynomial condition. Presence of transcendental functions (e.g., sin(dy/dx), e^(d²y/dx²), log(d³y/dx³)) containing derivatives is the primary cause.
✅ Correct Approach:
Before determining the degree, one must first ascertain if the given differential equation is a polynomial in its derivatives. This means no derivative term should be present inside a transcendental function (like sin, cos, tan, log, exponential) or have fractional/negative powers that cannot be cleared by raising both sides to an appropriate integer power. If the equation is not a polynomial in its derivatives, then its degree is undefined. If it is a polynomial, then identify the highest order derivative and its highest power to find the degree.
📝 Examples:
❌ Wrong:
Given ODE: sin(d2y/dx2) + (dy/dx)3 = x
Student's thought process: The highest order derivative is d2y/dx2. Its power is 1. So, order = 2, degree = 1.
Wrong Answer: Degree = 1
✅ Correct:
Given ODE: sin(d2y/dx2) + (dy/dx)3 = x
This equation cannot be expressed as a polynomial in its derivatives because of the sin(d2y/dx2) term. A derivative (d2y/dx2) is inside a transcendental function (sine).
Therefore, the order is 2, but the degree is undefined.
💡 Prevention Tips:
  • Understand the fundamental definition: The degree of an ODE is defined only if it is a polynomial in its derivatives.
  • Identify non-polynomial forms: Be vigilant for derivatives appearing as arguments of transcendental functions (e.g., sin(y'), cos(y''), e^(y'''), log(y'''')). In such cases, the degree is undefined.
  • CBSE Specific: Questions involving such cases are common to test conceptual clarity. Always double-check before stating the degree.
  • JEE Specific: This concept is equally important for JEE as it tests fundamental understanding, often in multiple-choice questions.
CBSE_12th
Minor Conceptual

❌ Incorrectly identifying the Degree when the ODE is not a polynomial in derivatives

Students often directly determine the degree of a differential equation without first ensuring that the equation is a polynomial in its derivatives. This commonly occurs when the equation involves radicals or fractional powers of derivatives, leading to an incorrect degree or stating it as 'undefined' prematurely.
💭 Why This Happens:
This conceptual mistake stems from an incomplete understanding of the definition of the degree of a differential equation. For the degree to be defined, the differential equation must be expressible as a polynomial in all its derivatives. Students frequently overlook this crucial pre-condition and proceed to identify the highest power of the highest order derivative without necessary algebraic manipulations.
✅ Correct Approach:
Before determining the degree, always ensure the given differential equation is free from radicals, fractional powers, or negative powers involving derivatives. Algebraically manipulate the equation (e.g., squaring both sides, raising to a suitable power) to eliminate these. Once the equation is a polynomial in its derivatives, then identify the highest power of the highest order derivative to find the degree.
📝 Examples:
❌ Wrong:
Consider the differential equation:
$$left(frac{d^2y}{dx^2}
ight)^{3/2} = 4 + frac{dy}{dx}$$
A common mistake is to identify the order as 2 (correct) and the degree as 3/2 (incorrect).
✅ Correct:
For the equation:
$$left(frac{d^2y}{dx^2}
ight)^{3/2} = 4 + frac{dy}{dx}$$
To find the degree, first eliminate the fractional power by squaring both sides:
$$left(left(frac{d^2y}{dx^2}
ight)^{3/2}
ight)^2 = left(4 + frac{dy}{dx}
ight)^2$$
$$left(frac{d^2y}{dx^2}
ight)^3 = 16 + 8frac{dy}{dx} + left(frac{dy}{dx}
ight)^2$$
Now, the equation is a polynomial in its derivatives.
The order is 2 (due to $$frac{d^2y}{dx^2}$$).
The degree is 3 (the power of the highest order derivative, $$frac{d^2y}{dx^2}$$).
💡 Prevention Tips:
  • Always Simplify First: Before finding the degree, check if the ODE contains any radicals or fractional/negative powers of derivatives.
  • Rationalize: Perform algebraic operations (like squaring, cubing, etc.) to make the equation a polynomial in derivatives.
  • Recall Definition: Remind yourself that the degree is the power of the highest order derivative after the equation is made polynomial in derivatives.
  • JEE Advanced Note: While simple for CBSE, in JEE, questions might involve functions like $$sin(frac{dy}{dx})$$ or $$e^{frac{d^2y}{dx^2}}$$, where the degree is genuinely undefined as they cannot be expressed as polynomials in derivatives. But for radicals/fractional powers, simplification is key.
CBSE_12th
Minor Conceptual

❌ Incorrectly assigning a degree to a non-polynomial differential equation

Students often attempt to determine the 'degree' of a differential equation even when it cannot be expressed as a polynomial in its derivatives. This occurs when derivatives are present inside transcendental functions (e.g., sin(y'), e^(y'')) or have fractional/negative powers, leading to an incorrect numerical value for the degree or an oversight that the degree is undefined.
💭 Why This Happens:
This mistake stems from a superficial understanding of the definition of 'degree'. Many students remember to find the highest power of the highest order derivative but forget the crucial prerequisite: the differential equation must be a polynomial in all its derivatives. Haste during exams and a lack of careful inspection for transcendental terms or non-integer powers of derivatives contribute significantly to this error.
✅ Correct Approach:
To correctly determine the order and degree of an ODE:
📝 Examples:
❌ Wrong: 
y''' + cos(y') + x = 0
Wrong Approach: "The highest order derivative is y''', its power is 1. So, order = 3, degree = 1."
✅ Correct: 
y''' + cos(y') + x = 0
Correct Approach:
  • The highest order derivative is y''', so the order is 3.
  • However, the term cos(y') involves a derivative (y') inside a transcendental function (cosine).
  • Therefore, this differential equation cannot be expressed as a polynomial in its derivatives.
  • Hence, the degree is undefined.
💡 Prevention Tips:
  • Master the Definition: Emphasize that degree is defined only if the ODE is a polynomial in its derivatives. This is a critical JEE concept.
  • Scrutinize Terms: Always check for derivatives within transcendental functions (sin, cos, tan, log, e^, etc.) or derivatives raised to fractional or negative powers. If found, the degree is undefined.
  • Avoid Manipulation: Do not try to 'rationalize' or 'simplify' terms involving derivatives in transcendental functions; such manipulation usually alters the equation or is not permitted for degree determination.
JEE_Advanced
Minor Calculation

❌ Ignoring Fractional Powers/Radicals when Determining Degree

A common minor mistake is incorrectly identifying the degree of a differential equation when it contains fractional powers or radicals involving derivatives. Students often pick the power of the highest order derivative as it initially appears, without first converting the equation into a polynomial form with respect to its derivatives.
💭 Why This Happens:
This error stems from a misunderstanding of the strict definition of 'degree'. The degree is defined only when the differential equation can be expressed as a polynomial in its derivatives. Haste in solving and a lack of careful algebraic manipulation to clear fractional exponents or radicals often lead to this mistake. It's easy to overlook this step, especially under exam pressure.
✅ Correct Approach:
To correctly find the degree of a differential equation, first ensure it is free from radicals and fractional powers of all derivatives. This often requires algebraic manipulation such as squaring, cubing, or raising both sides of the equation to an appropriate integral power. Once the equation is expressed as a polynomial in its derivatives, the degree is the highest power of the highest order derivative. The order is simply the highest order derivative present, irrespective of its form.
📝 Examples:
❌ Wrong: 
Wrong Approach:
For the ODE: $ left( frac{d^2y}{dx^2} 

ight)^{1/2} + frac{dy}{dx} = x $
Students might identify:
Order = 2 (due to $frac{d^2y}{dx^2}$)
Degree = 1/2 (incorrectly taking the power of $frac{d^2y}{dx^2}$ as it is)
✅ Correct: 
Correct Approach:
For the ODE: $ left( frac{d^2y}{dx^2} 

ight)^{1/2} + frac{dy}{dx} = x $
1. Identify the Order: The highest derivative is $frac{d^2y}{dx^2}$, so the Order = 2.
2. To find the Degree, eliminate the fractional power:
   $ left( frac{d^2y}{dx^2} 

ight)^{1/2} = x - frac{dy}{dx} $
   Squaring both sides to remove the fractional exponent:
   $ frac{d^2y}{dx^2} = left( x - frac{dy}{dx} 

ight)^2 $
   $ frac{d^2y}{dx^2} = x^2 - 2xfrac{dy}{dx} + left(frac{dy}{dx}

ight)^2 $
   Now, the equation is a polynomial in its derivatives. The highest order derivative is $frac{d^2y}{dx^2}$, and its power is 1.
   Therefore, the Degree = 1.
💡 Prevention Tips:
  • Thoroughly check the ODE's form: Before stating the degree, always inspect the equation for any fractional powers or radicals involving derivatives.
  • Algebraic Manipulation: Be prepared to perform necessary algebraic steps (like raising both sides to an appropriate power) to transform the ODE into a polynomial in its derivatives.
  • Definition Recall: Reiterate the definition of degree: 'The power of the highest order derivative after the differential equation is made free from radicals and fractional powers of all derivatives.'
JEE_Advanced
Minor Formula

❌ Ignoring Simplification Before Determining Degree

Students frequently correctly identify the order of a differential equation but falter in determining its degree when the equation is not presented as a polynomial in its derivatives. They often directly identify the highest power of the highest order derivative, overlooking the crucial step of converting the ODE into a polynomial form by clearing radicals or fractional powers involving derivatives.
💭 Why This Happens:
This error primarily stems from an incomplete understanding of the precise definition of 'degree'. The degree is rigorously defined as the power of the highest order derivative *after* the differential equation has been made free from radicals and fractions involving differential coefficients, essentially transforming it into a polynomial in all its derivatives. Students often skip this vital algebraic simplification, leading to incorrect degree assignment.
✅ Correct Approach:
To accurately determine the order and degree of an ODE (especially for JEE Advanced):
  • First, determine the order by identifying the highest order derivative present in the equation.
  • Next, if the equation contains radicals or fractional powers of derivatives, algebraically clear them (e.g., by isolating radical terms and raising both sides to appropriate integer powers) until the equation becomes a polynomial in all its derivatives.
  • Once in polynomial form, the degree is the power of the highest order derivative. Important: If the differential equation cannot be expressed as a polynomial in its derivatives (e.g., $ sin(y'') $ or $ e^{y'} $ terms), its degree is undefined.
📝 Examples:
❌ Wrong:
Consider the ODE: $ sqrt{1 + (frac{dy}{dx})^2} = frac{d^2y}{dx^2} $.
Mistake: Identifying Order = 2, Degree = 1 (by simply looking at the power of $ frac{d^2y}{dx^2} $). This ignores the radical term.
✅ Correct:
For the ODE: $ sqrt{1 + (frac{dy}{dx})^2} = frac{d^2y}{dx^2} $.
  1. The highest order derivative is $ frac{d^2y}{dx^2} $. So, Order = 2.
  2. To find the degree, we must first make it a polynomial in derivatives. Square both sides to eliminate the radical:
    $ 1 + (frac{dy}{dx})^2 = (frac{d^2y}{dx^2})^2 $
  3. Now, the equation is a polynomial in its derivatives. The highest order derivative is $ frac{d^2y}{dx^2} $, and its power is 2.
Therefore, Order = 2, Degree = 2.
💡 Prevention Tips:
  • Systematic Simplification: Before concluding the degree, always take an extra step to ensure the ODE is free from radicals and fractional powers of derivatives.
  • Know the 'Undefined' Cases: Be aware that if derivatives appear inside transcendental functions (like sin, cos, tan, log, e^x), the degree is undefined.
  • JEE Advanced Focus: Questions in JEE Advanced often include such 'tricky' forms to test a thorough understanding of definitions. Practice diverse problems involving these scenarios.
JEE_Advanced
Minor Unit Conversion

❌ Incorrect Determination of Degree Due to Non-Polynomial Form

Students frequently make errors in determining the degree of an Ordinary Differential Equation (ODE) when the equation is not a polynomial in its derivatives. They often overlook the essential step of making the equation free from radicals or fractional powers of the derivatives before identifying the degree.
💭 Why This Happens:
This mistake occurs because students sometimes rush to apply the definition of degree (highest power of the highest order derivative) without ensuring the ODE is in a proper polynomial form. They forget that the degree is defined only when the ODE can be expressed as a polynomial in its derivatives.
✅ Correct Approach:
To correctly find the degree of an ODE:
  • First, identify the order of the ODE (the order of the highest derivative present).
  • Next, eliminate any radicals or fractional powers of the derivatives by squaring, cubing, or raising to an appropriate power on both sides of the equation until all derivatives have integer powers.
  • Only then, identify the degree, which is the highest power of the highest order derivative after the equation has been made a polynomial in its derivatives.
JEE Advanced Tip: Be especially vigilant for ODEs involving square roots, cube roots, or fractional exponents of derivatives.
📝 Examples:
❌ Wrong:
Consider the ODE:
dy/dx + (d^2y/dx^2)^(1/2) = 0
A common incorrect approach is to identify the highest order derivative as d^2y/dx^2, and then simply state its power as 1/2, thus incorrectly concluding the degree is 1/2.
✅ Correct:
For the same ODE:
dy/dx + (d^2y/dx^2)^(1/2) = 0
1. The highest order derivative is d^2y/dx^2, so the order is 2.
2. To make it a polynomial, isolate the term with the fractional power:
(d^2y/dx^2)^(1/2) = -dy/dx
3. Square both sides to eliminate the fractional power:
d^2y/dx^2 = (-dy/dx)^2
d^2y/dx^2 = (dy/dx)^2
4. Now, the equation is a polynomial in its derivatives. The highest order derivative is d^2y/dx^2, and its power is 1.
Therefore, the degree is 1.
💡 Prevention Tips:
  • Always Check: Before stating the degree, ensure the ODE is free of radicals and fractional powers of derivatives.
  • Isolate terms with fractional powers before raising to an appropriate power to simplify the process.
  • Remember, order is always an integer, and degree (when defined) is also always a positive integer.
  • Practice problems specifically involving non-polynomial forms to solidify your understanding.
JEE_Advanced
Minor Sign Error

❌ <strong>Incorrect Sign Handling During Rationalization of Fractional Powers</strong>

Students often make algebraic errors when isolating terms with fractional powers or radicals, particularly with negative signs during rearrangement. This can lead to an incorrect intermediate equation or confusion, even if the final order and degree might coincidentally be correct, indicating a flawed algebraic process.
💭 Why This Happens:

  • Hasty Rearrangement: Overlooking negative signs when moving terms across equality.

  • Power Rule Confusion: Misunderstanding `(-A)^n` behavior (e.g., `(-A)^2 = A^2` vs `(-A)^3 = -A^3`).

  • Lack of Attention: Minor oversight of negative signs during simplification.

✅ Correct Approach:

  • Isolate Carefully: Isolate radical/fractional power terms, meticulously checking signs.

  • Apply Power: Raise both sides to the appropriate power, recalling `(-A)^n` rules.

  • Verify Form: Ensure the equation is a polynomial in derivatives for degree determination.

📝 Examples:
❌ Wrong:
Consider the equation:
`y''' + (y'')^{1/2} = 2x`

Wrong Step: A student incorrectly writes `(y'')^{1/2} = -(y''' - 2x)` instead of `2x - y'''` due to a sign error during rearrangement.

Squaring both sides yields: `y'' = (-(y''' - 2x))^2`

Which simplifies to: `y'' = (y''' - 2x)^2`

Expanding: `y'' = (y''')^2 - 4xy''' + 4x^2`

Note: While the final order (3) and degree (2) are coincidentally correct in this specific instance, this intermediate sign error reveals a deficiency in algebraic precision.
✅ Correct:
For the same equation:
`y''' + (y'')^{1/2} = 2x`


  1. Isolate the fractional power term: `(y'')^{1/2} = 2x - y'''`

  2. Square both sides to remove the radical: `y'' = (2x - y''')^2`

  3. Expand the right side to ensure polynomial form: `y'' = 4x^2 - 4xy''' + (y''')^2`


This equation is a polynomial in derivatives.

Highest order derivative is `y'''` (third order). So, Order = 3.

The power of the highest order derivative `y'''` is 2. So, Degree = 2.
💡 Prevention Tips:

  • Step-by-Step Isolation: Isolate terms meticulously, double-checking signs at each step of rearrangement.

  • Parentheses: Use parentheses for clarity, especially when handling negative expressions or multi-term exponents.

  • JEE Precision: Precision in all algebraic steps is crucial for JEE Advanced, as minor errors can cascade into larger issues in more complex problems.

JEE_Advanced
Minor Approximation

❌ Incorrect Degree Determination due to Unresolved Radicals/Fractional Powers

Students frequently make an error in determining the degree of an ordinary differential equation (ODE) when it is not presented as a polynomial in its derivatives. They often overlook or fail to eliminate radicals or fractional powers involving derivatives before identifying the highest power of the highest order derivative.
💭 Why This Happens:
This mistake stems from an incomplete understanding of the precise definition of 'degree'. The degree of an ODE is defined only when the equation can be expressed as a polynomial in all its derivatives. Students often rush to identify the highest power without first ensuring this condition is met, especially in JEE Advanced where such algebraic manipulation is expected.
✅ Correct Approach:
To correctly find the degree of an ODE, follow these steps:
1. First, identify the order of the differential equation, which is the highest order derivative present.
2. Before finding the degree, ensure the differential equation is free from radicals and fractional powers with respect to all derivatives. This usually involves isolating the radical/fractional term and raising both sides of the equation to a suitable power until all derivatives have integer exponents.
3. Once the equation is a polynomial in its derivatives, the degree is the highest power of the highest order derivative.
📝 Examples:
❌ Wrong:
Consider the ODE:
(d²y/dx²)³ = (dy/dx)² + √(d³y/dx³)
A common incorrect approach is to state that the order is 3 (from d³y/dx³) and the degree is 1 (the power of d³y/dx³ under the square root, without clearing it).
✅ Correct:
For the same ODE: (d²y/dx²)³ = (dy/dx)² + √(d³y/dx³)
1. Isolate the radical term:
(d²y/dx²)³ - (dy/dx)² = √(d³y/dx³)
2. Eliminate the radical by squaring both sides:
((d²y/dx²)³ - (dy/dx)²)² = (d³y/dx³)
3. Now the equation is a polynomial in its derivatives. The highest order derivative is d³y/dx³, so the Order = 3.
4. The power of this highest order derivative (d³y/dx³) is 1.
5. Therefore, the Degree = 1.
💡 Prevention Tips:
  • Understand the Definition: The degree is only defined if the ODE is a polynomial in its derivatives. This is a critical condition for both CBSE and JEE Advanced.
  • Systematic Simplification: Always eliminate radicals or fractional powers of derivatives by algebraic manipulation (e.g., squaring, cubing) before determining the degree.
  • Practice: Work through problems involving various forms of ODEs to reinforce this concept and build confidence.
JEE_Advanced
Important Calculation

❌ <strong>Incorrect Degree Determination with Fractional Powers/Radicals</strong>

Students often incorrectly determine the degree of a differential equation when it contains fractional powers or radicals involving derivatives. The degree is defined only when the ODE is a polynomial in its derivatives. Failing to clear fractional powers or radicals before identifying the highest power of the highest order derivative leads to an erroneous degree.
💭 Why This Happens:
  • Misunderstanding Degree Definition: The primary reason is not fully grasping that the degree is defined only when the differential equation is a polynomial in its derivatives.
  • Hasty Calculation: Students often rush to identify the highest power without first simplifying the equation into a polynomial form.
  • Algebraic Errors: Mistakes while squaring, cubing, or raising both sides to an appropriate power to eliminate fractions or radicals.
✅ Correct Approach:
  1. Identify the Order: Find the highest order derivative present in the equation. This directly gives the order of the ODE.
  2. Rationalize/Clear Fractional Powers: Remove all fractional powers and radicals involving derivatives by raising both sides of the equation to an appropriate integer power. This step is crucial to ensure the equation becomes a polynomial in its derivatives.
  3. Identify the Degree: Once the equation is a polynomial in derivatives, the power of the highest order derivative (identified in step 1) determines the degree. If the equation cannot be expressed as a polynomial in its derivatives (e.g., involving sin(dy/dx) or e^(d2y/dx2)), its degree is undefined.

JEE Advanced vs. CBSE: Both syllabi expect this understanding, but JEE Advanced frequently presents questions requiring more complex algebraic manipulations to test this concept thoroughly.

📝 Examples:
❌ Wrong:
Consider the differential equation: 
(d2y/dx2)3/2 = d3y/dx3

Incorrect approach: Students often identify the order as 3 but incorrectly state the degree as 3/2, or conclude it's undefined without proper algebraic manipulation.

✅ Correct:
For the equation: 
(d2y/dx2)3/2 = d3y/dx3
  1. Order: The highest order derivative is d3y/dx3. Thus, the Order is 3.
  2. Rationalize: To clear the fractional power (3/2), square both sides of the equation: 
    [(d2y/dx2)3/2]2 = (d3y/dx3)2
    This simplifies to: 
    (d2y/dx2)3 = (d3y/dx3)2
    Now, the equation is a polynomial in its derivatives.
  3. Degree: The highest order derivative is d3y/dx3, and its power in the polynomial form is 2. Therefore, the Degree is 2.
💡 Prevention Tips:
  • Master the Definitions: Thoroughly understand the precise definitions of order and degree, especially the condition that the differential equation must be a polynomial in its derivatives for the degree to be defined.
  • Systematic Approach: Always follow the steps: first identify the order, then rationalize to clear all fractional powers or radicals involving derivatives, and finally determine the degree.
  • Practice Algebraic Manipulation: Regularly practice simplifying expressions involving exponents and radicals to avoid errors during rationalization.
JEE_Advanced
Important Formula

❌ Incorrect Degree Calculation Due to Non-Polynomial Form or Radicals

Students frequently miscalculate the degree of an Ordinary Differential Equation (ODE) by not ensuring the equation is first a polynomial in its derivatives. This mistake is common when derivatives appear under fractional powers, roots, or as arguments of transcendental functions.
💭 Why This Happens:
This error often stems from a fundamental misunderstanding that the degree of a differential equation is defined only when it is a polynomial in all its derivatives. Hasty identification of the highest power without first eliminating radicals or fractions, or confusing the power with the defined degree, are common culprits.
✅ Correct Approach:
To correctly determine the order and degree of an ODE for JEE Advanced:
  1. Identify the highest order derivative present in the equation. This directly gives the order.
  2. Crucially, ensure the differential equation is a polynomial in all its derivatives. This means no derivative should be inside a root, fractional power, or transcendental function (e.g., sin, cos, log, exponential).
  3. If the equation is not a polynomial in its derivatives, algebraically manipulate it (e.g., squaring both sides, cubing both sides, clearing denominators) to eliminate any radicals or fractions involving the derivatives.
  4. Once the equation is in a polynomial form in its derivatives, the power of the highest order derivative gives its degree.
  5. Warning: If, after all possible algebraic manipulations, the equation cannot be expressed as a polynomial in its derivatives (e.g., sin(y'), e^(y'')), then the degree is undefined, though the order will still exist.
📝 Examples:
❌ Wrong:

Determine the order and degree of:

$left( frac{d^2y}{dx^2}
ight)^{3/2} = 4 + frac{dy}{dx}$

Wrong Approach:

  • Highest order derivative: $frac{d^2y}{dx^2}$. So, order = 2.
  • Power of highest order derivative: $3/2$. So, degree = $3/2$.

(Incorrect because the equation is not a polynomial in derivatives due to the fractional power.)

✅ Correct:

Determine the order and degree of:

$left( frac{d^2y}{dx^2}
ight)^{3/2} = 4 + frac{dy}{dx}$

Correct Approach:

  1. Identify the highest order derivative: $frac{d^2y}{dx^2}$. Hence, Order = 2.
  2. To find the degree, the equation must be free from fractional powers (radicals) involving derivatives. Square both sides to remove the $3/2$ power:
    3. $left( left( frac{d^2y}{dx^2}
    ight)^{3/2}
    ight)^2 = left( 4 + frac{dy}{dx}
    ight)^2$
    $left( frac{d^2y}{dx^2}
    ight)^3 = left( 4 + frac{dy}{dx}
    ight)^2$
  3. Now, the equation is a polynomial in its derivatives. The highest order derivative is $frac{d^2y}{dx^2}$, and its power is 3.
  4. Therefore, Degree = 3.
💡 Prevention Tips:
  • Always confirm the ODE is a polynomial in all its derivatives before stating the degree.
  • Algebraically manipulate the equation to eliminate radicals and fractional powers involving derivatives (e.g., squaring, cubing, multiplying by LCM of denominators).
  • If a derivative is within a transcendental function (e.g., sin(y'), log(y''), e^(y''')), the degree is undefined, though the order still exists. This is a frequent JEE Advanced trap.
  • CBSE vs JEE: While CBSE often presents simpler forms, JEE Advanced rigorously tests algebraic manipulation and a thorough understanding of when the degree is undefined. Be prepared for complex expressions!
JEE_Advanced
Important Unit Conversion

❌ Incorrect Degree Determination due to Non-Polynomial Form or Fractional Powers of Derivatives

A common mistake in ODEs, particularly for JEE Advanced, is incorrectly determining the degree of a differential equation. Students often overlook the crucial condition that the degree is only defined if the differential equation can be expressed as a polynomial in its derivatives. They may directly pick the power of the highest order derivative even when the equation is not in polynomial form, or if derivatives appear inside functions like trigonometric, exponential, or fractional powers.
💭 Why This Happens:
This error stems from an incomplete understanding of the definition of the degree of a differential equation. Students correctly identify the order (highest order derivative present) but fail to preprocess the equation to a polynomial form in derivatives before determining the degree. Confusion arises when derivatives are under a radical sign or raised to fractional powers, or when they are arguments of transcendental functions. Note: Unit conversion is not applicable to the 'order' and 'degree' of a differential equation, as these are dimensionless properties.
✅ Correct Approach:
To correctly determine the degree of an ODE:
📝 Examples:
❌ Wrong:

Consider the equation: 1 + (dy/dx)^2 = (d^2y/dx^2)^(1/3)

Students often identify the order as 2 and then try to take the power of (d^2y/dx^2), stating the degree as 1/3. This is incorrect.

✅ Correct:

For the equation: 1 + (dy/dx)^2 = (d^2y/dx^2)^(1/3)

  1. Order: The highest derivative is d^2y/dx^2, so the order is 2.
  2. Manipulate for Degree: To make it a polynomial in derivatives, cube both sides:
    [1 + (dy/dx)^2]^3 = [(d^2y/dx^2)^(1/3)]^3
    [1 + (dy/dx)^2]^3 = d^2y/dx^2
  3. Identify Degree: Now, the equation is a polynomial in derivatives. The highest order derivative is d^2y/dx^2, and its power is 1. Therefore, the degree is 1.
💡 Prevention Tips:
  • Memorize Definitions: Clearly understand that degree is defined only for equations that are polynomials in derivatives.
  • Always Pre-process: Before determining the degree, always check if there are any fractional powers or radicals involving derivatives. Clear them by raising both sides to appropriate powers.
  • Beware of Transcendental Functions: If derivatives are inside functions like sin(y''), e^(y'), or log(y'''), the degree is not defined.
  • Practice JEE Advanced Level Problems: Work through problems that specifically test the understanding of degree definition in complex forms.
JEE_Advanced
Important Sign Error

❌ <span style='color: red;'>Misinterpreting Negative Exponents or Coefficients for Degree Calculation</span>

Students often make algebraic sign errors when converting a differential equation into a polynomial in its derivatives, especially when dealing with negative exponents or terms requiring rearrangement. This can lead to incorrect clearing of radicals or fractions, and consequently, an erroneous determination of the degree of the ODE.
💭 Why This Happens:
  • Algebraic Slips: Incorrectly applying rules for negative exponents (e.g., confusing a-n = 1/an with a-n = -an) or mishandling signs during squaring/cubing terms.
  • Premature Degree Assessment: Attempting to determine the degree before the differential equation is truly in a polynomial form in terms of its derivatives.
✅ Correct Approach:
To correctly determine the degree of an ODE, follow these steps:
  1. Transform to Polynomial Form: Systematically eliminate all fractional and negative exponents from the derivatives. This involves isolating terms and raising both sides of the equation to appropriate integer powers. Ensure all derivatives are in the numerator.
  2. Identify Order: The order is the highest order derivative present in the equation after it has been simplified.
  3. Identify Degree: Once the equation is a polynomial in its derivatives, the degree is the highest power of the highest order derivative. Be meticulous with all algebraic signs during this process.
📝 Examples:
❌ Wrong:
Consider the ODE: (d2y/dx2)-1 + (dy/dx)2 = 5x
Incorrect approach: A student might mistakenly state, "The d2y/dx2 term has an exponent of -1, so the degree is -1 or undefined due to the negative sign." This directly misunderstands the definition of degree and the necessary algebraic transformations.
✅ Correct:
For the ODE: (d2y/dx2)-1 + (dy/dx)2 = 5x
  1. Rewrite the negative exponent: 1 / (d2y/dx2) + (dy/dx)2 = 5x
  2. Clear the fraction (multiply by d2y/dx2): 1 + (d2y/dx2) * (dy/dx)2 = 5x * (d2y/dx2)
  3. Rearrange into polynomial form: (d2y/dx2) * ( (dy/dx)2 - 5x ) + 1 = 0
In this equation, the order is 2 (due to d2y/dx2), and the degree is 1 (the highest power of the highest derivative).
💡 Prevention Tips:
  • Systematic Conversion: Always follow a systematic process to eliminate fractional and negative exponents from derivatives before assessing order and degree.
  • Algebraic Accuracy: Pay close attention to algebraic signs and rules, especially when transposing terms or raising expressions to powers.
  • Verify Polynomial Form: For JEE Advanced, ensure the equation is genuinely a polynomial in all its derivatives before concluding on the degree. If terms like sin(y') or ey'' remain, the degree is undefined.
JEE_Advanced
Important Approximation

❌ Ignoring Radical/Fractional Powers or Transcendental Functions in Derivatives for Degree Calculation

Students frequently make errors in determining the 'degree' of a differential equation by:
  • Failing to eliminate radicals or fractional powers involving derivatives before identifying the highest power of the highest order derivative.
  • Incorrectly assigning a degree when the equation contains transcendental functions (e.g., sin, cos, log, ex) whose arguments are derivatives. In such cases, the degree is undefined, a critical point often overlooked in JEE Advanced.
💭 Why This Happens:
This common mistake stems from an incomplete understanding of the precise definition of 'degree'. Students often rush, applying the power of the highest order derivative directly without ensuring the differential equation is a polynomial in its derivatives. The strict conditions for 'degree' (polynomial form, no fractional powers, no transcendental functions of derivatives) are frequently forgotten or misapplied.
✅ Correct Approach:
To correctly determine the order and degree of an ODE:
  • 1. Identify the Order: The order is always the highest order of the derivative present in the equation. This is generally straightforward.
  • 2. Check for Transcendental Functions: If any derivative (of any order) appears as the argument of a transcendental function (e.g., sin(dy/dx), log(d²y/dx²), edy/dx), then the ODE cannot be expressed as a polynomial in its derivatives. In this scenario, the degree is undefined.
  • 3. Clear Radicals/Fractional Powers (if applicable): If the degree is not undefined, make the equation free from all radicals and fractional powers involving derivatives by raising both sides to appropriate integral powers. This transforms the equation into a polynomial in terms of its derivatives.
  • 4. Identify the Degree: Once the equation is a polynomial in its derivatives, the degree is the power of the highest order derivative.
📝 Examples:
❌ Wrong:
Consider the differential equation: dy/dx + √(d²y/dx²) = x
A student might incorrectly identify:
Order = 2 (Correct)
Degree = 1 (Incorrect, as the radical is not cleared)
✅ Correct:
For the differential equation: dy/dx + √(d²y/dx²) = x
  • Step 1: Identify Order
    The highest derivative is d²y/dx². So, Order = 2.
  • Step 2: Clear Radicals for Degree
    Rearrange the equation to isolate the radical: √(d²y/dx²) = x - dy/dx
    Square both sides to remove the radical: d²y/dx² = (x - dy/dx)²
    Expand the right side (optional for degree): d²y/dx² = x² - 2x(dy/dx) + (dy/dx)²
  • Step 3: Identify Degree
    Now the equation is a polynomial in derivatives. The highest order derivative is d²y/dx², and its power is 1. Therefore, Degree = 1.

JEE Advanced Tip: For sin(dy/dx) + d²y/dx² = 0, the degree is Undefined because of sin(dy/dx).
💡 Prevention Tips:
  • Strict Definition Recall: Always remember that 'degree' is defined only if the differential equation can be expressed as a polynomial in its derivatives.
  • Systematic Elimination: Before determining the degree, always check for and eliminate all fractional powers and radicals involving derivatives.
  • Transcendental Function Check: Be extremely vigilant for derivatives appearing inside transcendental functions (sin, cos, log, etc.). If they do, the degree is immediately undefined. This is a common trick question in JEE Advanced.
  • Practice: Solve a variety of problems, specifically those involving radicals and transcendental functions, to internalize the correct procedure.
JEE_Advanced
Important Other

❌ Incorrectly Determining Degree Due to Non-Polynomial Form of ODE

Many students rush to determine the order and degree of a differential equation without first ensuring it is expressed as a polynomial in its derivatives. The degree of a differential equation is defined only when the equation can be written as a polynomial in derivatives. If there are fractional powers or radicals involving derivatives, the degree cannot be directly identified.
💭 Why This Happens:
This mistake commonly occurs due to overlooking the fundamental definition of the degree of a differential equation. Students often jump to identify the highest power of the highest order derivative without clearing radicals or fractional exponents, which are often used in JEE Advanced problems to test this specific understanding. Lack of careful manipulation and simplification before applying the definition is the primary cause.
✅ Correct Approach:
Before determining the degree, the given differential equation must be converted into a polynomial in its derivatives. This involves eliminating all fractional powers and radicals involving the derivatives by suitable algebraic manipulations (e.g., squaring both sides, cubing both sides, etc.). Once the equation is free of such terms and is a polynomial in all its derivatives, then identify the highest order derivative and its highest power (exponent) to find the degree.
📝 Examples:
❌ Wrong:
Consider the equation: $(frac{d^2y}{dx^2})^{1/2} = frac{dy}{dx} + x$
Incorrect thought process: The highest order derivative is $d^2y/dx^2$, so order = 2. Its power is 1/2, so degree = 1/2. This is incorrect as the equation is not a polynomial in derivatives.
✅ Correct:
For the same equation: $(frac{d^2y}{dx^2})^{1/2} = frac{dy}{dx} + x$
Correct approach: To make it a polynomial in derivatives, square both sides:
$frac{d^2y}{dx^2} = (frac{dy}{dx} + x)^2$
$frac{d^2y}{dx^2} = (frac{dy}{dx})^2 + 2xfrac{dy}{dx} + x^2$
Now the equation is a polynomial in its derivatives. The highest order derivative is $d^2y/dx^2$, so the order is 2. The power of this highest order derivative is 1, so the degree is 1.
💡 Prevention Tips:
  • Always check: Ensure the differential equation is a polynomial in its derivatives before determining the degree.
  • Clear radicals/fractions: Perform necessary algebraic manipulations (squaring, cubing, raising to an appropriate power) to remove all fractional powers or radicals involving derivatives.
  • JEE Advanced Focus: Be extra vigilant in JEE Advanced, as questions often deliberately include such forms to trick students.
  • Practice: Solve multiple problems involving equations with fractional powers or radicals to build confidence in correctly identifying order and degree.
JEE_Advanced
Important Conceptual

❌ Incorrectly Determining Degree for Non-Polynomial Differential Equations

A frequent conceptual error is attempting to find the degree of a differential equation even when it cannot be expressed as a polynomial in its derivatives. This leads students to provide an incorrect numerical value for the degree, whereas the correct answer should be 'not defined'. This mistake is particularly common in JEE Advanced.
💭 Why This Happens:
This error stems from an incomplete understanding of the fundamental condition for the existence of a degree. Students often rush to identify the highest power of the highest order derivative without first verifying if the differential equation can be algebraically cleared of radicals and fractions, and crucially, if all derivatives are outside transcendental functions (like sin, cos, tan, log, e^x).
✅ Correct Approach:
  1. Identify the Order: First, determine the order of the differential equation, which is simply the order of the highest derivative present in the equation.
  2. Check Polynomial Form: Next, critically examine if the differential equation can be written as a polynomial in all its derivatives. This means no derivative should appear inside a trigonometric function (e.g., sin(y')), exponential function (e^(y'')), logarithmic function (log(y')), or have fractional/negative powers (after clearing radicals/fractions).
  3. Determine Degree (if defined): If the equation *can* be made polynomial in its derivatives, then the degree is the power of the highest order derivative when the equation is free from radicals and fractional powers with respect to the derivatives.
  4. Degree Not Defined: If the equation *cannot* be made polynomial in its derivatives, then the degree is not defined.
📝 Examples:
❌ Wrong:
Equation: e^(d^2y/dx^2) + (dy/dx)^3 = x
Wrong Approach: Students might correctly identify the order as 2, then incorrectly try to assign a degree, perhaps 1 (from the implicit power of e^(y'')) or 3 (from (dy/dx)^3).
✅ Correct:
Equation: e^(d^2y/dx^2) + (dy/dx)^3 = x
Correct Approach: The term e^(d^2y/dx^2) involves the second derivative within an exponential function. This means the differential equation cannot be expressed as a polynomial in its derivatives.
Therefore, its Order is 2, but its Degree is not defined.
💡 Prevention Tips:
  • Prioritize the polynomial condition for degree: Always check this crucial step before assigning a numerical value to the degree.
  • Practice differentiating between expressions like sin(x) * (dy/dx) (where degree is defined, as dy/dx is multiplied by sin(x), not inside it) and sin(dy/dx) (where degree is not defined).
  • For CBSE, this concept is often tested directly. For JEE Advanced, it can be a quick check or part of a multi-conceptual problem.
JEE_Advanced
Important Unit Conversion

❌ Incorrect Determination of Degree for Non-Polynomial Differential Equations

Students frequently make errors in finding the degree of a differential equation when it is not a polynomial in its derivatives. This often involves neglecting to clear fractional powers, roots, or derivatives in denominators, leading to an incorrect or undefined degree. Note: Unit conversion is not applicable when determining the order and degree of ordinary differential equations. These are purely mathematical properties of the equation's structure.
💭 Why This Happens:
  • Lack of understanding that the degree is defined only for differential equations that are polynomials in their derivatives.
  • Hasty calculation without proper algebraic manipulation (e.g., clearing fractional powers or roots).
  • Confusing the power of the highest order derivative with the degree before making the equation polynomial in its derivatives.
✅ Correct Approach:

1. Identify the highest order derivative in the equation. This directly gives the order.

2. To find the degree, first ensure the differential equation is free from radicals (roots) and fractions involving derivatives. This means rewriting the equation as a polynomial in its derivatives. For JEE Main, this step is crucial and often tested.

3. Once in polynomial form, the degree is the power of the highest order derivative. If it cannot be expressed as a polynomial in its derivatives (e.g., terms like `sin(dy/dx)` or `e^(d^2y/dx^2)` exist), the degree is undefined.

📝 Examples:
❌ Wrong:

Consider the equation: dy/dx + (d^2y/dx^2)^(3/2) = 0

A common mistake is to directly state:

  • Order = 2 (correct)
  • Degree = 3/2 (incorrect, degree must be a non-negative integer for it to be defined).
✅ Correct:

For the same equation: dy/dx + (d^2y/dx^2)^(3/2) = 0

1. Rewrite to isolate the term with the fractional power:

(d^2y/dx^2)^(3/2) = -dy/dx

2. Square both sides to eliminate the fractional power:

((d^2y/dx^2)^(3/2))^2 = (-dy/dx)^2

(d^2y/dx^2)^3 = (dy/dx)^2

3. Now, the equation is a polynomial in its derivatives.

  • Order = 2 (highest derivative is d^2y/dx^2)
  • Degree = 3 (power of the highest order derivative, which is d^2y/dx^2, once in polynomial form)
💡 Prevention Tips:
  • Always clear fractions and radicals: Before determining the degree, ensure the equation is free from fractional powers and roots of derivatives. This is a fundamental step for both CBSE and JEE Main.
  • Check for polynomial form: The degree is only defined if the differential equation can be expressed as a polynomial in its derivatives. If terms like sin(dy/dx), log(d^2y/dx^2), or e^(d^3y/dx^3) are present, the degree is undefined.
  • Identify order first: The order is the highest derivative present. The degree is the power of that specific highest order derivative once the equation is in polynomial form.
  • Practice algebraic manipulation: Be proficient in isolating and squaring/cubing terms to eliminate roots and fractional powers accurately.
JEE_Main
Important Other

❌ Incorrect Degree Determination due to Radicals or Fractional Powers

Students often fail to simplify the differential equation into a polynomial form in its derivatives before determining its degree. This leads to incorrect identification of the degree, especially when the equation involves radicals or fractional powers of derivatives.
💭 Why This Happens:

This mistake commonly occurs because students overlook the fundamental definition of the 'degree' of a differential equation. The degree is defined only when the differential equation is expressible as a polynomial in its derivatives. If radicals or fractional powers of derivatives are present, these must be eliminated first by appropriate algebraic manipulations (like squaring, cubing, etc.). Rushing through the problem without proper simplification is a primary reason.

✅ Correct Approach:

To correctly find the degree:

  1. First, determine the order of the differential equation, which is the order of the highest derivative present.
  2. Next, manipulate the equation algebraically to make it free from radicals and fractional powers of all derivatives. The equation must be expressed as a polynomial in its derivatives.
  3. Once in polynomial form, the degree is the highest power of the highest order derivative present in the equation.

JEE Tip: Always prioritize converting to polynomial form. If an ODE cannot be expressed as a polynomial in its derivatives, its degree is undefined.

📝 Examples:
❌ Wrong:

Consider the differential equation:
1 + (dy/dx)^2 = (d^2y/dx^2)^(1/3)
Incorrect approach: A student might directly state the order as 2 and the degree as 1/3, seeing the power of the highest derivative.

✅ Correct:

For the same differential equation:
1 + (dy/dx)^2 = (d^2y/dx^2)^(1/3)
Correct approach:

  1. Identify the highest order derivative: d^2y/dx^2. So, the Order = 2.
  2. To eliminate the fractional power (1/3), cube both sides of the equation:
    (1 + (dy/dx)^2)^3 = ((d^2y/dx^2)^(1/3))^3
    (1 + (dy/dx)^2)^3 = (d^2y/dx^2)^1
  3. Now, the equation is a polynomial in its derivatives. The highest order derivative is d^2y/dx^2, and its power is 1.
  4. Therefore, the Degree = 1.
💡 Prevention Tips:
  • Understand Definitions: Thoroughly grasp that degree is defined only for equations that are polynomials in derivatives.
  • Always Simplify: Before finding the degree, ensure the equation is cleared of all radicals and fractional powers of derivatives.
  • Double-Check: After simplifying, re-identify the highest order derivative and then its highest power.
  • Practice: Work through problems involving various forms of ODEs, including those with implicit fractional powers (like square roots).
JEE_Main
Important Approximation

❌ Incorrectly Determining Degree Due to Non-Polynomial Form

Students frequently make errors in determining the degree of a differential equation by failing to first express it as a polynomial in its derivatives. They might mistakenly take the power of the highest order derivative even when it's under a radical, a fractional exponent, or within a transcendental function.
💭 Why This Happens:
This mistake stems from a superficial understanding of the definition of 'degree'. The formal definition requires the differential equation to be a polynomial in its derivatives. Students often overlook this crucial condition, or they rush, neglecting the necessary algebraic manipulations to remove radicals or fractional/negative powers involving derivatives.
✅ Correct Approach:
To correctly determine the order and degree of a differential equation:
  • Order: The order is simply the highest order of the derivative present in the equation. This is generally straightforward.
  • Degree:
    1. First, identify the order.
    2. Next, algebraically manipulate the equation to eliminate any radicals, fractional powers, or negative powers of derivatives. This often involves raising both sides to a suitable power.
    3. Ensure the equation is now expressible as a polynomial in all its derivatives.
    4. Once in this polynomial form, the degree is the highest power of the highest order derivative (identified in step 1).
    5. Important: If the differential equation cannot be expressed as a polynomial in its derivatives (e.g., contains terms like sin(y'), e^(y''), log(y''')), then the degree is undefined.
📝 Examples:
❌ Wrong:

Consider the equation: d2y/dx2 + (dy/dx)1/2 = 0

Wrong Approach: Students might identify order = 2 and then incorrectly state degree = 1, assuming the power of d2y/dx2 is 1 without further manipulation.

✅ Correct:

Consider the equation: d2y/dx2 + (dy/dx)1/2 = 0

Correct Approach:

  1. Order: The highest order derivative is d2y/dx2, so the Order = 2.
  2. Degree:
    Rearrange the equation to isolate the fractional power: d2y/dx2 = - (dy/dx)1/2
    Square both sides to eliminate the fractional power: (d2y/dx2)2 = (dy/dx)
    Now, the equation is (d2y/dx2)2 - (dy/dx) = 0, which is a polynomial in its derivatives.
    The highest power of the highest order derivative (d2y/dx2) is 2. Therefore, the Degree = 2.
💡 Prevention Tips:
  • JEE Specific: Always be cautious with equations involving radicals, fractional powers, or transcendental functions of derivatives. These are common traps set in JEE Main questions.
  • Check for Polynomial Form: Before declaring the degree, ensure the differential equation is free from radicals, fractional/negative powers, and functions (like sin, cos, log, e) of any derivative.
  • If Undefined: Remember that if an equation contains terms like sin(y') or e^(y''), its degree is undefined, even though its order can still be determined.
  • Practice Algebraic Manipulation: Develop strong skills in rearranging equations to clear radicals and fractional powers.
JEE_Main
Important Sign Error

❌ Sign Errors in Eliminating Radicals/Fractional Powers for Degree Determination

Students frequently make algebraic sign errors when manipulating differential equations to convert them into a polynomial form of derivatives, which is a prerequisite for determining the degree. Specifically, errors occur when raising terms involving negative signs to certain powers to clear radicals or fractional exponents. This can lead to an incorrectly transformed equation, making the subsequent determination of the degree unreliable or wrong.
💭 Why This Happens:
This error primarily stems from a lack of careful application of basic algebraic rules for exponents involving negative bases. Students often confuse `(-A)^2` with `-(A^2)` or `(-A)^3` with `A^3`. When an entire term with a negative sign is raised to an even power, the result is positive (e.g., `(-X)^2 = X^2`). Conversely, when raised to an odd power, the negative sign persists (e.g., `(-X)^3 = -X^3`). Rushing through algebraic steps and overlooking these fundamental rules are common causes.
✅ Correct Approach:
Always ensure the given differential equation is first expressed as a polynomial in its derivatives before attempting to find its degree. When eliminating radicals or fractional powers, apply exponent rules meticulously, especially when negative signs are involved. Remember that the degree is the highest power of the highest order derivative *after* the equation has been made a polynomial in derivatives. Double-check all algebraic manipulations, particularly those involving signs.
📝 Examples:
❌ Wrong:
Consider the ODE: (d2y/dx2)1/2 = - (dy/dx)
A student might incorrectly square both sides as:
d2y/dx2 = - (dy/dx)2
Here, the student incorrectly assumes that the negative sign on the RHS should remain outside the squared term, mistaking `(-A)^2` for `-(A^2)`. While the highest order derivative is d2y/dx2 with power 1 (Degree = 1), the algebraic transformation is flawed, indicating a fundamental sign error.
✅ Correct:
For the same ODE: (d2y/dx2)1/2 = - (dy/dx)
To eliminate the fractional power, we must square both sides correctly:
( (d2y/dx2)1/2 )2 = ( - (dy/dx) )2
d2y/dx2 = (dy/dx)2
Now the equation is a polynomial in its derivatives. The highest order derivative is d2y/dx2, and its power in this polynomial form is 1.
Therefore, the Order is 2, and the Degree is 1.
💡 Prevention Tips:
  • Master Exponent Rules: Pay special attention to how negative signs behave when terms are raised to even or odd powers (e.g., `(-A)^2 = A^2`, `(-A)^3 = -A^3`).
  • Step-by-Step Simplification: Avoid rushing. Break down the process of clearing radicals or fractional exponents into smaller, manageable steps.
  • Verify Polynomial Form: Always ensure the final form of the ODE is truly a polynomial in all its derivatives before concluding the degree. Any remaining fractional or negative powers indicate an incomplete transformation.
  • Practice Algebraic Manipulations: Regular practice with algebraic identities and transformations involving signs will build confidence and accuracy.
CBSE_12th
Important Formula

❌ Incorrectly Determining Degree for Non-Polynomial Differential Equations

Students often make mistakes when calculating the degree of a differential equation (ODE) if it is not expressed as a polynomial in its derivatives. The degree is defined only when the ODE can be written as a polynomial in all its derivatives. If the equation contains terms like sin(y'), e^(y''), or fractional/radical powers of derivatives, students might incorrectly state a degree or proceed without proper simplification.
💭 Why This Happens:
This mistake stems from a misunderstanding of the strict definition of degree. Students might:
  • Forget to eliminate radicals or fractional powers of derivatives.
  • Try to assign a degree even when transcendental functions (like trigonometric, exponential, or logarithmic) of derivatives are present.
  • Confuse the exponent of the highest order derivative with the degree without ensuring the polynomial form.
✅ Correct Approach:
To correctly determine the degree of an ODE:
  1. First, find the Order: Identify the highest order derivative present in the equation. This determines the order.
  2. For Degree, Check Polynomial Form: Ensure the ODE is expressible as a polynomial in all its derivatives. This means no fractional/radical powers of derivatives, and no transcendental functions of derivatives.
  3. Clear Radicals/Fractions: If radical signs or fractional powers of derivatives are present, square/cube/raise to appropriate power on both sides to clear them until all derivatives have integer powers.
  4. Identify Exponent: Once the ODE is polynomial in its derivatives, the degree is the highest power of the highest order derivative.
  5. Important JEE Note: If an ODE cannot be expressed as a polynomial in its derivatives (e.g., sin(dydx) term), its degree is not defined.
📝 Examples:
❌ Wrong:
Consider the ODE: (d2ydx2)3/2=dydx+x
Student's mistake: Order = 2, Degree = 3/2 (or 3, if only looking at the numerator of the power).
✅ Correct:
For the ODE: (d2ydx2)3/2=dydx+x
1. The highest order derivative is d2ydx2, so Order = 2.
2. To find the degree, we must make it a polynomial in derivatives. Square both sides:
[(d2ydx2)3/2]2=(dydx+x)2
(d2ydx2)3=(dydx+x)2
Now, the equation is a polynomial in its derivatives. The highest order derivative is d2ydx2, and its power is 3. So, Degree = 3.
💡 Prevention Tips:
  • Memorize Definitions Precisely: Understand that degree requires the ODE to be a polynomial in derivatives.
  • Always Simplify First: Before determining the degree, ensure all fractional or radical powers of derivatives are eliminated by suitable algebraic manipulation (e.g., raising both sides to a power).
  • Check for Transcendental Functions: If the ODE involves transcendental functions (like sin, cos, tan, log, e^x) of any derivative, the degree is undefined.
  • Practice Variety: Solve problems involving different forms of ODEs, including those where the degree is undefined or requires initial simplification.
  • Double-Check: After finding the order and degree, quickly re-verify your steps, especially the polynomial form for degree calculation.
JEE_Main
Important Calculation

❌ Miscalculating Degree Due to Fractional Powers

Students often correctly identify the order of a differential equation but make errors in determining its degree, particularly when the equation contains fractional powers (radicals) of derivative terms. They might incorrectly state the degree without first converting the equation into a polynomial in its derivatives.
💭 Why This Happens:
This error stems from an incomplete understanding of the definition of 'degree' for a differential equation. The degree is defined only if the differential equation can be expressed as a polynomial in its derivatives. If fractional powers exist, they must be eliminated by algebraic manipulation before determining the degree. Failure to do so leads to an incorrect calculation of the degree.
✅ Correct Approach:

  1. Identify the order of the differential equation, which is the order of the highest derivative present.

  2. To find the degree, first ensure the equation is free from radicals or fractional powers of all derivatives. Manipulate the equation algebraically (e.g., by raising both sides to a suitable power) until all derivatives appear with integer powers.

  3. Once the equation is a polynomial in its derivatives, the degree is the highest power of the highest order derivative.

📝 Examples:
❌ Wrong:
Consider the equation:
$(d^2y/dx^2)^{1/2} = dy/dx + x$
  • Incorrect Approach: Order = 2, Degree = 1/2 (from $d^2y/dx^2$).
✅ Correct:
For the same equation:
$(d^2y/dx^2)^{1/2} = dy/dx + x$
  • Correct Approach: Square both sides to eliminate the fractional power:
    $d^2y/dx^2 = (dy/dx + x)^2$
    Expanding the RHS:
    $d^2y/dx^2 = (dy/dx)^2 + 2x(dy/dx) + x^2$
    Now, the equation is a polynomial in its derivatives.
    The highest order derivative is $d^2y/dx^2$ (order 2).
    Its power is 1.
    Therefore, Order = 2, Degree = 1.
💡 Prevention Tips:

  • Algebraic Simplification: Always eliminate radicals or fractional powers involving derivatives first. This is crucial before attempting to find the degree.

  • Polynomial Form Rule: Remember that the degree is defined only for differential equations that can be expressed as a polynomial in their derivatives.

  • JEE Relevance: This specific type of algebraic manipulation and subsequent degree determination is a common trap in competitive exams like JEE Main.

JEE_Main
Important Conceptual

❌ Incorrectly Determining Degree for Non-Polynomial ODEs

Students often make mistakes when determining the degree of a differential equation, especially when the equation is not a polynomial in its derivatives. The mistake arises from directly identifying the power of the highest order derivative without first ensuring that the equation is free from fractional powers, radicals, or transcendental functions of the derivatives.
💭 Why This Happens:
This error stems from a lack of complete conceptual understanding of the definition of 'degree'. The degree is defined as the power of the highest order derivative after the differential equation has been made free from radicals and fractional powers of derivatives. Students often overlook this crucial prerequisite, or they attempt to find the degree when it's undefined due to transcendental functions of derivatives.
✅ Correct Approach:
To correctly determine the degree of an Ordinary Differential Equation (ODE):
  • First, identify the order, which is the order of the highest derivative present in the equation.
  • Next, ensure the equation is a polynomial in all its derivatives. This means eliminating any fractional powers, radicals (square roots, cube roots, etc.), or negative powers of derivatives. This is usually done by isolating and raising both sides to appropriate powers.
  • If, after this simplification, the equation cannot be expressed as a polynomial in its derivatives (e.g., sin(dy/dx) or e^(d2y/dx2) is present), then its degree is undefined.
  • If it is a polynomial, the degree is the power of the highest order derivative.

JEE Main Tip: Questions on degree often involve such transformations to test conceptual clarity.
📝 Examples:
❌ Wrong:
Consider the differential equation: y'' + (y')^(1/2) = 0
Wrong Approach: Students might incorrectly identify the highest order derivative as y'' (order 2) and simply state its power, 1, as the degree.
✅ Correct:
For the differential equation: y'' + (y')^(1/2) = 0
  • Step 1: Isolate the term with the fractional power: y'' = -(y')^(1/2)
  • Step 2: Square both sides to remove the fractional power: (y'')^2 = (-(y')^(1/2))^2 which simplifies to (y'')^2 = y'.
  • Step 3: Now the equation is a polynomial in derivatives. The highest order derivative is y'' (order 2).
  • Step 4: Its power is 2. Therefore, the order is 2 and the degree is 2.
💡 Prevention Tips:
  • Always Simplify First: Before determining the degree, ensure the ODE is free from radicals and fractional powers of derivatives.
  • Check for Transcendental Functions: If derivatives are inside trigonometric, exponential, or logarithmic functions (e.g., sin(y')), the degree is undefined.
  • Practice Variety: Solve numerous problems involving different forms of ODEs to reinforce the simplification process.
  • Review Definitions: Revisit the precise definitions of order and degree, paying special attention to the conditions for degree.
JEE_Main
Important Approximation

❌ Ignoring Non-Polynomial Form or Fractional Powers in Derivatives for Degree

Students frequently make an 'approximation understanding' error by directly identifying the power of the highest order derivative as the degree, without first ensuring the Differential Equation is a polynomial in its derivatives. This oversight leads to two common scenarios of error:
  • Incorrectly assigning a degree when transcendental functions (like sin, cos, e^x, log) involve a derivative as their argument (e.g., sin(dy/dx)).
  • Failing to clear fractional or negative powers of derivatives (e.g., (d²y/dx²)^(3/2)) before determining the degree, which must always be a positive integer.
💭 Why This Happens:
This mistake stems from a mechanical application of the 'highest power of the highest derivative' rule without understanding its crucial prerequisite: the differential equation must be expressible as a polynomial in all its derivatives. Students often confuse the power of the dependent variable 'y' with the power of the derivative, or simply overlook the algebraic manipulation required to remove non-integer powers or identify transcendental forms.
✅ Correct Approach:

To correctly determine the order and degree:

  1. Determine the Order: Identify the highest order derivative present in the equation. This is always defined.
  2. Determine the Degree:
    • First, check if the equation is a polynomial in its derivatives. This means no derivative should be inside a transcendental function (e.g., sin(y'), e^(y''), log(y''')). If such terms exist, the degree is not defined.
    • Next, ensure all derivatives have positive integer powers. If fractional or negative powers of derivatives are present, clear them by appropriate algebraic manipulation (e.g., squaring both sides, cubing both sides, multiplying by a term with a negative power) until all derivatives have integer exponents. This step might change the form of other terms in the equation.
    • Once the equation is a polynomial in its derivatives, the degree is the highest power of the highest order derivative.
📝 Examples:
❌ Wrong:

Equation: dy/dx + sin(d²y/dx²) = 0

Wrong identification:
Order = 2
Degree = 1 (assuming sin(d²y/dx²) has power 1 or ignoring it completely for degree).

Equation: (d²y/dx²)^(3/2) = dy/dx + y

Wrong identification:
Order = 2
Degree = 3/2 (directly taking the fractional power).

✅ Correct:

For Equation: dy/dx + sin(d²y/dx²) = 0

  • Order: The highest derivative is d²y/dx², so the order is 2.
  • Degree: Since the term sin(d²y/dx²) involves a derivative as an argument of a sine function, the equation is not a polynomial in its derivatives. Therefore, the degree is Not Defined.

For Equation: (d²y/dx²)^(3/2) = dy/dx + y

  • Step 1: Clear fractional powers. Square both sides:
    [(d²y/dx²)^(3/2)]² = (dy/dx + y)²
    (d²y/dx²)^3 = (dy/dx + y)²
  • Step 2: Identify highest order derivative and its power.
    The highest order derivative is d²y/dx². Its power in the polynomial form is 3.
  • Order: 2
  • Degree: 3
💡 Prevention Tips:
  • Always double-check the definition: Degree is only defined for differential equations that are polynomials in their derivatives. This is a critical point for CBSE 12th Exams.
  • Beware of Transcendental Functions: If derivatives are arguments of functions like sin(), cos(), tan(), e^(), log(), the degree is immediately not defined.
  • Eliminate Fractional/Negative Powers: Before determining degree, perform necessary algebraic operations (like squaring, cubing, or multiplying to clear denominators) to ensure all derivatives have positive integer powers.
  • Practice: Work through various examples, especially those with non-polynomial forms or fractional powers, to solidify your understanding.
CBSE_12th
Important Other

❌ Incorrectly determining Degree when ODE is not a Polynomial in Derivatives

Many students rush to determine the degree of a differential equation without first ensuring it is expressible as a polynomial in its derivatives. This often leads to errors when the equation contains fractional powers or radicals involving derivative terms.
💭 Why This Happens:
This mistake occurs primarily due to a misunderstanding or oversight of the crucial condition for defining the degree: the differential equation must be a polynomial in its derivatives. Students often correctly identify the order (highest derivative present) but then directly pick the power of that highest derivative without eliminating radicals or fractional powers, which makes the equation non-polynomial in derivatives.
✅ Correct Approach:
To correctly determine the degree of an ordinary differential equation, follow these steps:
1. Identify the Order: Find the highest order derivative present in the equation. This determines the 'order'.
2. Eliminate Radicals/Fractions: If the equation contains radicals or fractional powers involving any derivative, rewrite the equation to clear these. This often involves raising both sides of the equation to a suitable power.
3. Ensure Polynomial Form: The equation must be expressed as a polynomial in its derivatives.
4. Determine Degree: After ensuring it's in polynomial form, the highest power of the highest order derivative determines the 'degree'.
📝 Examples:
❌ Wrong:
Consider the ODE: (dy/dx) + (d²y/dx²)^(3/2) = 0
Some students might state:
Order = 2 (due to d²y/dx²)
Degree = 3/2 (taking the power of the highest order derivative directly)
✅ Correct:
For the same ODE: (dy/dx) + (d²y/dx²)^(3/2) = 0
1. Isolate the term with the fractional power: (d²y/dx²)^(3/2) = - (dy/dx)
2. Square both sides to eliminate the fractional power: (d²y/dx²)^3 = (-dy/dx)²
3. Now, the equation is (d²y/dx²)^3 = (dy/dx)², which is a polynomial in its derivatives.
4. The highest order derivative is d²y/dx², and its highest power is 3.
Therefore:
Order = 2
Degree = 3
💡 Prevention Tips:
  • Always Simplify First: Before anything else, manipulate the ODE to remove all radical signs and fractional powers involving derivatives.
  • Check for 'Polynomial in Derivatives': Mentally (or physically) verify if the ODE looks like a polynomial where the variables are the derivatives.
  • CBSE Specific: For CBSE exams, this is a common trick question. Always prioritize the 'polynomial in derivatives' condition before determining degree.
  • Practice Questions: Solve various problems, especially those with radicals or complex powers, to solidify your understanding.
CBSE_12th
Important Sign Error

❌ Sign Error in Handling Negative Exponents in Derivatives

Students often misinterpret a negative exponent in a derivative term, such as (dy/dx)-1, as a negative coefficient or a negative sign of the derivative, e.g., -(dy/dx). This fundamental algebraic error leads to an incorrect polynomial form of the differential equation, consequently affecting the determination of its degree.
💭 Why This Happens:
This error primarily stems from a conceptual misunderstanding of exponent rules. Students may confuse `a-n = 1/an` with `-an`, especially when under exam pressure or if their algebraic foundations are weak. They might rush through the rearrangement process, overlooking the reciprocal nature of negative exponents.
✅ Correct Approach:
To correctly determine the degree, the ODE must first be expressed as a polynomial in derivatives. When encountering a derivative term with a negative exponent, always convert it into its reciprocal form. Then, clear any denominators to ensure all derivatives appear with positive integer powers. Only after the equation is free from fractions and radicals with respect to derivatives can the degree be identified as the highest power of the highest order derivative.
📝 Examples:
❌ Wrong:
Consider the ODE:
d²y/dx² + (dy/dx)⁻¹ = 0
Wrong approach: Student assumes (dy/dx)⁻¹ = -(dy/dx).
This leads to: d²y/dx² - dy/dx = 0.
Here, the order is 2 and the degree is incorrectly identified as 1.
✅ Correct:
For the same ODE:
d²y/dx² + (dy/dx)⁻¹ = 0
Correct approach: Understand that (dy/dx)⁻¹ = 1/(dy/dx).
The equation becomes: d²y/dx² + 1/(dy/dx) = 0
Multiply by (dy/dx) to clear the denominator:
(d²y/dx²)(dy/dx) + 1 = 0
This is now a polynomial in derivatives. The highest order derivative is d²y/dx², and its power is 1. The order is 2, and the correct degree is 1.
💡 Prevention Tips:
  • Review Exponent Rules: Solidify your understanding of basic exponent rules, especially `a-n = 1/an`.
  • Step-by-Step Simplification: Always convert negative exponents to reciprocals and clear denominators systematically before determining the degree.
  • Verify Polynomial Form: Ensure the ODE is truly a polynomial in derivatives (no fractional or negative powers, no transcendental functions of derivatives) before identifying the degree.
  • JEE Main Specific: Such basic algebraic errors are common traps. Practice problems involving various forms of ODEs to reinforce correct manipulation.
JEE_Main
Important Unit Conversion

❌ Incorrectly Determining the Degree of an ODE

Students frequently make errors in determining the degree of a differential equation. This often happens due to a misunderstanding of the conditions required for a degree to be defined, specifically when the equation involves radicals, fractional powers of derivatives, or when derivatives appear inside non-polynomial (transcendental) functions.
💭 Why This Happens:
  • Lack of understanding that the degree is defined only when the differential equation can be expressed as a polynomial in terms of its derivatives.
  • Failing to eliminate radicals or fractional powers involving derivatives before identifying the degree.
  • Not recognizing that the degree is undefined if any derivative (e.g., dy/dx, d²y/dx²) is an argument of a transcendental function (like sin, cos, tan, log, e^).
  • Sometimes, confusing the power of a lower-order derivative with the power of the highest-order derivative after clearing fractions/radicals.
✅ Correct Approach:
  1. Identify the Order: First, determine the order of the ODE, which is the highest order of the derivative present in the equation.
  2. Clear Radicals/Fractional Powers (CBSE & JEE): If the differential equation contains radicals or fractional powers involving derivatives, manipulate the equation algebraically to eliminate them, ensuring the equation becomes a polynomial in terms of its derivatives. This usually involves raising both sides to appropriate integer powers.
  3. Check for Transcendental Functions (CBSE & JEE): Examine if any derivative appears as the argument of a transcendental function (e.g., sin(d²y/dx²), e^(dy/dx), log(y''')). If so, the degree is undefined, and no further steps are needed.
  4. Determine Degree: After successfully completing steps 2 and 3 (or if they were not applicable), the degree is the highest integral power of the highest-order derivative (identified in step 1) in the differential equation.
📝 Examples:
❌ Wrong:
Consider the ODE: (d³y/dx³)²/³ + d²y/dx² = y
A common mistake is to state the degree as 2/3 directly without clearing the fractional power, or to confuse it with the power of d²y/dx².
✅ Correct:
Consider the ODE: (d³y/dx³)²/³ + d²y/dx² = y
1. Order: The highest derivative is d³y/dx³, so the Order is 3.
2. Clear Fractional Power: Isolate the term with the fractional power:
(d³y/dx³)²/³ = y - d²y/dx²
Cube both sides to remove the denominator of the power:
((d³y/dx³)²/³ )³ = (y - d²y/dx²)³
(d³y/dx³)² = (y - d²y/dx²)³
3. Check for Transcendental Functions: No derivatives are inside transcendental functions.
4. Degree: The highest order derivative is d³y/dx³, and its power is 2.
Therefore, the Degree = 2.

Another example where degree is undefined:
For sin(d²y/dx²) + x(dy/dx) = 0, the Order is 2. However, since d²y/dx² is an argument of the sin() function, the equation is not a polynomial in its derivatives. Hence, the degree is undefined.
💡 Prevention Tips:
  • Tip 1 (CBSE & JEE): Always ensure the differential equation is free from radicals and fractional powers involving derivatives before determining the degree. This is a crucial algebraic manipulation step.
  • Tip 2 (CBSE & JEE): Memorize the condition for an undefined degree: if any derivative is an argument of a transcendental function (sin, cos, tan, log, e^, etc.), the degree is undefined.
  • Tip 3: Practice with a variety of ODEs, focusing on those with complex structures to build confidence in identifying both order and degree correctly.
  • Tip 4: Distinguish clearly between order (highest derivative) and degree (power of that highest derivative after polynomial form).
CBSE_12th
Important Formula

❌ Ignoring the 'Polynomial in Derivatives' Condition for Degree

A common mistake is incorrectly determining the degree of a differential equation when it cannot be expressed as a polynomial in its derivatives. Students often confuse the highest power of the highest order derivative with the actual degree, overlooking the fundamental condition that all derivatives must appear as integer powers and not be inside transcendental functions (like sin, cos, log, e^x) or have fractional/negative powers.
💭 Why This Happens:
This error stems from a partial understanding of the definition of 'degree'. While the order is straightforward (highest derivative present), the degree requires an additional check: the differential equation must be reducible to a polynomial form with respect to its derivatives. Students frequently skip this critical step, especially when dealing with radicals or functions of derivatives.
✅ Correct Approach:
Before attempting to find the degree, first ensure that the given differential equation is free from radicals involving derivatives and that no derivative is an argument of a transcendental function. If fractional powers exist, clear them by raising both sides to an appropriate integer power. If a derivative is within a function like sin(y'), e^(y''), or log(y'), the degree is undefined. Only after ensuring it's a polynomial in its derivatives, identify the highest power of the highest order derivative.
📝 Examples:
❌ Wrong:
Consider the equation: 
d2y/dx2 = (1 + (dy/dx)2)3/2

A common incorrect approach is to state: Order = 2, Degree = 1 (taking the power of the LHS) or Degree = 3/2 (directly using the fractional power).
✅ Correct:
For the equation: 
d2y/dx2 = (1 + (dy/dx)2)3/2


  1. Eliminate fractional powers: Square both sides to remove the (3/2) power.
    (d2y/dx2)2 = ((1 + (dy/dx)2)3/2)2

    This simplifies to:
    (d2y/dx2)2 = (1 + (dy/dx)2)3


  2. Check for polynomial form: The equation is now a polynomial in its derivatives.

  3. Identify Order: The highest order derivative is d2y/dx2. So, Order = 2.

  4. Identify Degree: The highest power of the highest order derivative (d2y/dx2) is 2. So, Degree = 2.

💡 Prevention Tips:

  • Master the Definition: Thoroughly understand that for degree, the ODE must be expressible as a polynomial in its derivatives.

  • Check for Radicals and Functions: Always check if derivatives are under radicals or inside transcendental functions (sin, cos, log, e^x). If so, try to clear radicals; if a derivative is an argument of a transcendental function, the degree is undefined.

  • Simplify First: Always simplify the differential equation to its polynomial form in derivatives before determining the degree.

  • Practice with Variety: Solve problems involving ODEs with fractional powers, negative powers, and derivatives within transcendental functions to solidify understanding.
CBSE_12th
Important Conceptual

❌ Incorrectly determining Degree when the ODE is not a polynomial in its derivatives

Students frequently attempt to find the degree of an ordinary differential equation (ODE) even when it cannot be expressed as a polynomial in its derivatives, leading to an incorrect or undefined degree. This is a crucial conceptual error.
💭 Why This Happens:
This mistake stems from a fundamental misunderstanding that the degree is defined only for ODEs that are polynomials in all their derivatives. Students often overlook derivatives involved within non-algebraic functions (like trigonometric, exponential, or logarithmic functions) or radical/fractional powers that cannot be eliminated, mistakenly assigning a degree.
✅ Correct Approach:
To correctly determine the degree of an ODE:
📝 Examples:
❌ Wrong:
Consider the ODE:
d²y/dx² + sin(dy/dx) = 0
A common mistake is to state the degree as 1, assuming the power of the highest derivative (d²y/dx²) determines it, ignoring the non-polynomial term.
✅ Correct:
For the ODE:
d²y/dx² + sin(dy/dx) = 0
  • Order: 2 (The highest order derivative is d²y/dx²).
  • Degree: Undefined. This is because the term sin(dy/dx) involves a derivative (dy/dx) inside a trigonometric function, making the entire equation non-polynomial in its derivatives. Hence, the degree cannot be determined.
💡 Prevention Tips:
  • Always check if the ODE is a polynomial in all its derivatives before attempting to find the degree.
  • Be vigilant for derivatives enclosed within functions like sin(y'), e^(y''), or log(y'''). In such cases, the degree is unequivocally undefined.
  • Remember that order is always defined (for a well-formed ODE), but degree may be undefined. This is a frequent trick question in exams.
CBSE_12th
Important Calculation

❌ Incorrectly determining the Degree when the ODE is not a polynomial in its derivatives.

Students frequently attempt to determine the degree of a differential equation without first checking if it can be expressed as a polynomial in its derivatives. The degree is strictly defined as the highest power of the highest order derivative only after the equation is free of radicals and fractional powers of the derivatives and is a polynomial in derivatives. If derivatives are present within non-polynomial functions (e.g., sin(y'), e^(y''), log(y''')), or under irreducible fractional/radical powers, the degree is undefined.
💭 Why This Happens:
  • Incomplete Understanding of Definition: Many students overlook the crucial condition that the equation must be a polynomial in its derivatives for the degree to be defined.
  • Hasty Application: Students often rush to identify the highest power without simplifying fractional powers or verifying the polynomial form.
  • Confusion with Order: While the order is always defined (the highest derivative present), the degree has specific prerequisites for its existence.
✅ Correct Approach:
  1. First, identify the order of the differential equation (the highest order derivative present).
  2. Next, critically examine if the differential equation can be written as a polynomial in its derivatives. This means no derivatives inside trigonometric functions (sin(y'), cos(y'')), exponential functions (e^(y'')), logarithmic functions (log(y''')), or under non-integer powers (e.g., √(dy/dx), (d²y/dx²)^(1/2)).
  3. If it can, ensure all derivatives have integer powers by clearing any radicals or fractional powers (e.g., by squaring both sides).
  4. Once the equation is a polynomial in its derivatives with integer powers, the power of the highest order derivative (identified in step 1) is its degree.
  5. If it cannot be expressed as a polynomial in its derivatives, then the degree is undefined.
📝 Examples:
❌ Wrong: 
Find the degree of:

y'' + sin(y') = 0

Student's thought process: Highest order derivative is y'' (order 2). Its power is 1. So, degree is 1.

Result: Degree = 1 (Incorrect)
✅ Correct: 
Determine the order and degree of:

y'' + sin(y') = 0

Step 1: Identify the highest order derivative. It is y''.

Step 2: The order of the differential equation is 2.

Step 3: Check for polynomial form. Here, sin(y') involves the first derivative inside a trigonometric function. This equation cannot be expressed as a polynomial in its derivatives.

Step 4: Therefore, the degree is undefined.



Consider another example (CBSE 12th typical):

(d²y/dx²)^(3/2) = dy/dx + x

Step 1: Clear fractional powers by squaring both sides: (d²y/dx²)^3 = (dy/dx + x)²

Step 2: Highest order derivative is d²y/dx² (order 2).

Step 3: The equation is now a polynomial in its derivatives.

Step 4: The power of the highest order derivative (d²y/dx²) is 3.

Result: Order = 2, Degree = 3.
💡 Prevention Tips:
  • Master the Definition: Always recall that degree is the power of the highest order derivative only after it's made a polynomial in derivatives and free of radicals/fractional powers.
  • Two-Step Verification: First, find the order. Second, check if the equation is a polynomial in derivatives. If not, degree is undefined.
  • Look for Red Flags: If you see derivatives inside sin, cos, tan, log, e^(...) functions, immediately suspect that the degree might be undefined.
  • Simplify Aggressively: For fractional or radical powers of derivatives, clear them by raising both sides of the equation to appropriate integer powers before determining the degree.
  • Practice Diverse Problems: Work through examples covering all scenarios, especially those with non-polynomial forms and fractional powers, to build confidence.
CBSE_12th
Critical Other

❌ Incorrectly Determining Degree when ODE is Not a Polynomial in Derivatives

A common and critical mistake is attempting to determine the degree of an ordinary differential equation (ODE) when it cannot be expressed as a polynomial in its derivatives. This often occurs when derivatives appear under radical signs, fractional powers, or as arguments of transcendental functions (e.g., trigonometric, exponential, or logarithmic functions).
💭 Why This Happens:
This error stems from an incomplete understanding of the fundamental definition of the 'degree' of a differential equation. Students frequently overlook the crucial prerequisite that the ODE must first be convertible into a polynomial form with respect to its derivatives. They directly try to identify the highest power of the highest order derivative without ensuring this essential condition is met.
✅ Correct Approach:
Before determining the degree, one must rigorously ensure that the given differential equation can be expressed as a polynomial in its derivatives. If derivatives are under radicals or fractional powers, algebraic manipulation (e.g., squaring both sides, raising to a certain power) is necessary to clear these. If derivatives appear as arguments inside transcendental functions (like sin(d²y/dx²), e^(dy/dx), log(d³y/dx³)), then the degree of the differential equation is not defined. For the degree to be defined, all derivatives must appear with integer powers. Once the equation is in polynomial form, the degree is the highest power of the highest order derivative.
📝 Examples:
❌ Wrong:

Consider the ODE: d²y/dx² + cos(dy/dx) = 0

Wrong Approach: Students might identify the order as 2 and then mistakenly state the degree as 1 (assuming cos() doesn't affect the power of dy/dx or d²y/dx²).

✅ Correct:

Consider the ODE: d²y/dx² + cos(dy/dx) = 0

Correct Approach:

  • Order: The highest order derivative is d²y/dx², so the order is 2.
  • Degree: The term cos(dy/dx) means the differential equation is not a polynomial in its derivatives (specifically dy/dx). Since dy/dx is an argument of a transcendental function, its degree is not defined.

Similarly, for (d³y/dx³)² = 1 + √(dy/dx), first square both sides to get (d³y/dx³)^4 = (1 + dy/dx), which is a polynomial. Order=3, Degree=4.

💡 Prevention Tips:
  • Master the Definition: Thoroughly understand that the degree is only defined for ODEs that can be expressed as a polynomial in their derivatives.
  • Check for Transcendental Functions: Always scrutinize if any derivative is an argument of a trigonometric, exponential, or logarithmic function. If so, the degree is undefined.
  • Clear Radicals/Fractions: If derivatives are under radicals or fractional powers, perform necessary algebraic manipulations to convert the ODE into a polynomial form *before* attempting to find the degree.
  • CBSE vs. JEE: This concept is equally crucial for both CBSE board exams and JEE. In CBSE, a clear explanation for undefined degree is expected.
CBSE_12th
Critical Conceptual

❌ Incorrect Degree: Non-Polynomial Forms & Radicals

Students fail to determine the degree of an Ordinary Differential Equation (ODE) by not ensuring it's a polynomial in its derivatives or by neglecting to clear radicals/fractions involving derivatives. This is a critical conceptual error for JEE Main.
💭 Why This Happens:
This stems from haste and a misunderstanding of the strict definition: for its degree to be defined, an ODE must be expressible as a polynomial in all its derivatives, free from radicals and fractional powers of derivatives.
✅ Correct Approach:

To correctly find the degree:

  1. First, identify the Order, which is the order of the highest derivative present in the equation.
  2. Clear Radicals/Fractions: Algebraically eliminate all radicals or fractional powers involving derivatives (e.g., by squaring both sides) until all derivatives have integer powers.
  3. Check Polynomial Form: After clearing radicals, ensure the equation is a polynomial in its derivatives. If terms like sin(y'), e^(y''), log(y'''), etc., are present, the equation is not a polynomial in its derivatives, and thus, its Degree is undefined.
  4. Identify Degree: If it is a polynomial in derivatives, the Degree is the power of the highest order derivative appearing in the equation.
📝 Examples:
❌ Wrong:
Students frequently state Degree = 1 for y'' + sin(y') = 0 (missing the undefined case). Similarly, for y = x(dy/dx) + √(1 + (dy/dx)²), they often incorrectly identify the degree as 1.
✅ Correct:
  • Case 1: Non-Polynomial Form
    For y'' + sin(y') = 0:
    Order = 2 (due to y''). However, the term sin(y') means the equation is not a polynomial in its derivatives. Hence, the Degree is undefined.
  • Case 2: Radicals/Fractions
    For y = x(dy/dx) + √(1 + (dy/dx)²):
    1. Isolate radical: y - x(dy/dx) = √(1 + (dy/dx)²)
    2. Square both sides: (y - x(dy/dx))² = 1 + (dy/dx)²
    3. Expand and rearrange: (x²-1)(dy/dx)² - 2xy(dy/dx) + y² - 1 = 0
    Now, the highest derivative is (dy/dx) (Order = 1). Its highest power is 2. So, the Degree = 2.
    JEE Tip: Failing to clear the radical first is a common trap.
💡 Prevention Tips:
  • Always verify the 'polynomial in derivatives' condition.
  • Eliminate any radicals or fractional powers involving derivatives algebraically.
  • If derivatives are arguments of non-algebraic functions (e.g., sin, log, exp), the degree is undefined.
  • JEE Main Focus: These conceptual traps are very common. Practice problems involving both 'undefined degree' and radical removal.
JEE_Main
Critical Approximation

❌ Incorrectly Determining Degree When ODE is Not a Polynomial in Derivatives

Students frequently make the critical error of assigning a degree to a differential equation without first verifying if it can be expressed as a polynomial in its derivatives. This means they incorrectly state the degree when the equation contains derivatives within transcendental functions (like sin, cos, log, e^), or when derivatives have fractional powers that cannot be cleared.
💭 Why This Happens:
This mistake stems from a fundamental misunderstanding of the strict definition of 'degree' for a differential equation. Students often jump directly to identifying the highest power of the highest order derivative, overlooking the prerequisite that the equation must be expressible as a polynomial in its derivatives. They may confuse the 'order' (which is always defined) with the 'degree' (which has specific conditions for its existence).
✅ Correct Approach:
To correctly determine the order and degree of a differential equation:
📝 Examples:
❌ Wrong:
Consider the differential equation: dy/dx + sin(d^2y/dx^2) = 0

Incorrect thinking: The highest order derivative is d^2y/dx^2, so the order is 2. Its power is 1, so the degree is 1. (This is wrong because of the sin function.)

✅ Correct:
For the equation: dy/dx + sin(d^2y/dx^2) = 0

  • Order: The highest order derivative present is d^2y/dx^2. Therefore, the order is 2.

  • Degree: The term sin(d^2y/dx^2) involves the second derivative as the argument of a sine function. This means the differential equation cannot be expressed as a polynomial in its derivatives. Consequently, the degree is not defined.

💡 Prevention Tips:

  • Strictly Adhere to the Definition: Remember, the degree is the highest power of the highest order derivative only after the equation has been made free from radicals and fractional powers of the derivatives, and can be written as a polynomial in the derivatives.

  • Identify Transcendental Functions: Always check if any derivative appears inside functions like sin, cos, tan, log, or as an exponent (e.g., e^(y'), e^(y'')). If so, the degree is not defined. This is a common trap in both CBSE and JEE.

  • Eliminate Radicals/Fractional Powers: If terms like sqrt(dy/dx) or (d^2y/dx^2)^(3/2) are present, raise both sides of the equation to an appropriate power (e.g., square or cube) to clear all fractional powers before attempting to determine the degree.

CBSE_12th
Critical Sign Error

❌ Sign Error in Squaring/Cubing to Remove Radicals for Degree Calculation

Students often make critical sign errors when manipulating differential equations to remove radicals or fractional powers involving derivatives. This typically occurs during the squaring or cubing of both sides, leading to an incorrect polynomial form of the ODE. A common pitfall is misinterpreting how negative signs are handled in squaring, e.g., assuming `(-A)^2` is `-A^2` or mishandling signs when rearranging terms, which directly impacts the correct identification of the degree of the differential equation.
💭 Why This Happens:
This error primarily stems from fundamental algebraic weaknesses, specifically in applying exponent rules to negative terms or expressions. Carelessness during equation manipulation, such as moving terms across the equality sign without changing their sign, also contributes. Additionally, a lack of clear understanding that the goal is to transform the ODE into a polynomial in terms of its derivatives, where coefficients (including negative ones) do not alter the degree of a term, can lead to confusion.
✅ Correct Approach:
To correctly determine the degree of an ODE, ensure it is first made free from radicals and fractional powers with respect to its derivatives. Follow these steps carefully:
  1. Isolate the term containing the radical or fractional power of a derivative.
  2. Raise both sides of the equation to an appropriate integer power to eliminate the radical/fractional power.
  3. Be meticulous with algebraic signs during this process. Remember that squaring a negative term yields a positive result, i.e., (-A)^2 = A^2.
  4. Once the equation is a polynomial in derivatives, identify the highest order derivative. Its power in this polynomial form is the degree.
📝 Examples:
❌ Wrong:
Consider the ODE:
dy/dx = -√(d²y/dx²)
A common sign error during squaring would be:
(dy/dx)² = -(d²y/dx²)
In this incorrect step, the student fails to correctly square the negative sign outside the radical, leading to a misrepresentation of the equation's polynomial form and potential confusion in identifying the degree of the highest order derivative.
✅ Correct:
For the ODE:
dy/dx = -√(d²y/dx²)
  1. Square both sides correctly: (dy/dx)² = (-√(d²y/dx²))²
  2. Simplify: (dy/dx)² = d²y/dx²
  3. Rearrange to standard form (optional for degree): d²y/dx² - (dy/dx)² = 0
Here, the highest order derivative is d²y/dx² (order 2). Its power in the polynomial form is 1.
Therefore, Order = 2, Degree = 1.

JEE/CBSE Note: The degree is only defined if the ODE can be expressed as a polynomial in derivatives.
💡 Prevention Tips:
  • Master Basic Algebra: Reinforce fundamental rules for exponents, especially `(-A)^n`.
  • Check All Steps: When eliminating radicals, double-check every sign change and squaring operation.
  • Understand the Definition: Always recall that the degree is the power of the highest order derivative *after* the ODE is a polynomial in derivatives.
  • Coefficient vs. Power: A negative coefficient (e.g., `-d²y/dx²`) does not change the power of the derivative (which is 1 in this case), it's just a constant multiplier.
CBSE_12th
Critical Unit Conversion

❌ <span style='color: #FF0000;'>Confusing Order and Degree, and Failing to Identify When Degree is Undefined</span>

Students frequently misidentify the order and degree of an Ordinary Differential Equation (ODE). A critical error of critical severity is attempting to assign a numerical value to the degree even when the ODE cannot be expressed as a polynomial in its derivatives, leading to an incorrect answer of '1', '2', etc., instead of correctly stating that the degree is undefined. Other errors include incorrectly identifying the highest derivative for order or failing to simplify the equation (e.g., removing radicals) before determining the degree.
💭 Why This Happens:
This mistake stems from a superficial understanding of the precise definitions of order and degree. Students often overlook the crucial condition that the degree is only defined if the ODE is a polynomial in all its derivatives. Hasty examination for terms like sin(y'), e^(y''), or log(y'), which prevent the equation from being a polynomial, is a common pitfall. Similarly, ignoring fractional powers or radicals over derivatives leads to incorrect degree determination.
✅ Correct Approach:
  1. Determine Order First: Identify the highest order derivative present in the ODE. The order of this derivative is the order of the ODE.
  2. Check for Polynomial Form (for Degree):
    • Ensure the equation is free from radicals or fractional powers of any derivative. If not, clear them by squaring/cubing both sides as necessary to obtain integer powers.
    • Scrutinize the equation for non-polynomial terms involving derivatives, such as sin(y'), cos(y''), e(y'), log(y'''), etc.
    • If such terms are present, the degree is undefined.
    • If the ODE can be expressed as a polynomial in its derivatives, the degree is the highest power of the highest order derivative.
📝 Examples:
❌ Wrong:
Consider the ODE: 
dy/dx + sin(d2y/dx2) = 0
Incorrect Approach: Order = 2, Degree = 1 (Student assumes degree is 1 because sin is raised to power 1, ignoring that sin is a function of the derivative).
✅ Correct:
Consider the ODE: 
dy/dx + sin(d2y/dx2) = 0
Correct Approach:
  • Order: The highest order derivative is d2y/dx2. Therefore, the order is 2.
  • Degree: The term sin(d2y/dx2) makes the ODE non-polynomial in its derivatives. Hence, the degree is undefined.
💡 Prevention Tips:
  • Master Definitions: Commit the precise definitions of order and degree, along with the conditions for degree existence, to memory.
  • Systematic Check: Always follow the two-step process: first order, then degree. For degree, always check for non-polynomial functions and fractional powers/radicals of derivatives.
  • CBSE vs. JEE: This concept is fundamental for both. While CBSE questions might be direct, JEE could present more complex forms requiring algebraic manipulation before determining the degree. Ensure your equation is in a form where all derivatives have integer powers before finding the degree.
  • Practice Diverse Problems: Work through examples including those with non-polynomial terms and fractional powers to solidify your understanding.
CBSE_12th
Critical Formula

❌ <span style='color: #FF0000;'>Incorrectly Determining the Degree When Undefined</span>

A critical mistake students often make is attempting to find the degree of an Ordinary Differential Equation (ODE) even when it is not a polynomial in its derivatives. They overlook the fundamental condition that for the degree to be defined, the ODE must first be expressible as a polynomial in all its derivatives (i.e., no derivatives within transcendental functions like sin, cos, log, e^x, or having fractional/radical powers that cannot be removed).
💭 Why This Happens:
This error stems from a rushed application of definitions without checking all prerequisites. Students often memorize that 'degree is the power of the highest order derivative' but forget the crucial preceding condition about the ODE being a polynomial in derivatives. They might also confuse 'power' with 'degree' or fail to rationalize fractional powers correctly (though the polynomial check is usually the primary issue in CBSE).
✅ Correct Approach:
To correctly determine the degree of an ODE:
  1. First, identify the order, which is the order of the highest derivative present in the equation.
  2. Next, check if the equation can be expressed as a polynomial in all its derivatives. This means no derivative should be inside a trigonometric function (e.g., sin(dy/dx)), exponential function (e^(d^2y/dx^2)), logarithmic function, or have non-integer powers that cannot be cleared by squaring/cubing both sides (i.e., it must be free from radicals and fractional powers in terms of derivatives).
  3. If it cannot be expressed as a polynomial in its derivatives, then the degree is undefined.
  4. If it can be expressed as a polynomial in derivatives, then the degree is the highest power of the highest order derivative present in the equation.
📝 Examples:
❌ Wrong:
Consider the ODE: e(d3y/dx3) + (d2y/dx2) = x
A common wrong answer for degree would be 1 (the power of the d2y/dx2 term). This ignores the exponential term involving the third derivative.
✅ Correct:
For the ODE: e(d3y/dx3) + (d2y/dx2) = x
  • Order: 3 (due to d3y/dx3).
  • Degree: Undefined. This is because the term e(d3y/dx3) is not a polynomial in derivatives. You cannot express this equation as a polynomial in d3y/dx3.

For comparison, consider: (d2y/dx2)3 + (dy/dx)2 + y = 0
  • Order: 2 (due to d2y/dx2).
  • Degree: 3 (the power of the highest order derivative, d2y/dx2).
💡 Prevention Tips:
  • Always Check First: Before assigning a degree, always verify if the ODE is a polynomial in its derivatives. This is the golden rule for CBSE and JEE.
  • Look for Transcendental Functions: Be vigilant for derivatives within sin(), cos(), tan(), log(), e(). If found, the degree is undefined.
  • Rationalize Powers: Ensure all derivatives have integer powers by squaring/cubing if necessary. If fractional powers remain after all possible rationalization, the degree is undefined.
  • Practice: Work through diverse examples to solidify this understanding, especially those with undefined degrees.
CBSE_12th
Critical Calculation

❌ Incorrectly Determining Degree When ODE is Not a Polynomial in Derivatives

Students frequently make critical errors in calculating the degree of an Ordinary Differential Equation (ODE) by not first ensuring that the equation is expressed as a polynomial in its derivatives. This commonly occurs when the ODE involves fractional powers (radicals) of derivatives or derivatives inside non-algebraic functions (like trigonometric, exponential, or logarithmic functions). They might either incorrectly assign a degree from the original form or state 'undefined' without proper steps.
💭 Why This Happens:
This mistake stems from a fundamental misunderstanding of the definition of 'degree' for an ODE. The degree is only defined if the differential equation can be written as a polynomial in its derivatives. Students often rush to find the highest power of the highest order derivative without clearing fractional powers or realizing that the equation is not a polynomial in derivatives at all. Lack of careful algebraic manipulation also contributes to this error.
✅ Correct Approach:

To correctly determine the degree of an ODE:

  1. Identify the Order: First, find the order of the differential equation, which is the highest order of the derivative present in the equation.
  2. Clear Fractional Powers/Radicals: If the ODE contains fractional powers or radicals involving derivatives, algebraically manipulate the equation to eliminate them. This typically involves isolating terms and raising both sides to appropriate integral powers until all derivative terms have integral exponents.
  3. Check for Polynomial Form: After clearing fractional powers, ensure the equation is now a polynomial in its derivatives. If any derivative (of any order) is part of a transcendental function (e.g., sin(dy/dx), e^(d²y/dx²), log(d³y/dx³)), then the degree is undefined.
  4. Determine Degree: Once the equation is a polynomial in its derivatives, the degree is the highest integral power of the highest order derivative found in step 1.
📝 Examples:
❌ Wrong:

Consider the ODE: (dy/dx)^(3/2) + (d²y/dx²)^2 = 0

Wrong Approach: Some students might incorrectly state the order as 2 and the degree as 2 (from (d²y/dx²)²) or even 3/2 (from (dy/dx)^(3/2)) directly without proper manipulation.

✅ Correct:

For the ODE: (dy/dx)^(3/2) + (d²y/dx²)^2 = 0

  1. Isolate the fractional power term: (dy/dx)^(3/2) = -(d²y/dx²)^2
  2. Clear the fractional power by squaring both sides: ((dy/dx)^(3/2))^2 = (-(d²y/dx²)^2)^2
    (dy/dx)³ = (d²y/dx²)⁴
  3. Now, the equation is a polynomial in its derivatives.
  4. Highest order derivative: d²y/dx²
  5. Order: 2
  6. Highest integral power of the highest order derivative: The power of (d²y/dx²) is 4.
  7. Therefore, Degree = 4.
💡 Prevention Tips:
  • Master the Definitions: Understand that degree is strictly defined only for ODEs that are polynomials in their derivatives.
  • Always Manipulate First: Before determining the degree, always clear radicals or fractional powers involving derivatives through appropriate algebraic steps.
  • Check for Transcendental Functions: If any derivative term is inside a sine, cosine, log, or exponential function, the degree is unequivocally undefined. (JEE vs CBSE: This point is critical for both, but more strictly tested in CBSE where direct application of the rule is common.)
  • Practice Algebraic Simplification: Strong algebraic skills are crucial for correctly transforming the ODE into a polynomial form.
CBSE_12th
Critical Other

❌ Incorrectly Determining Degree for Non-Polynomial Differential Equations

Students frequently attempt to assign a numerical degree to a differential equation even when it is not expressible as a polynomial in its derivatives, resulting in an incorrect integer value instead of the correct answer: undefined.
💭 Why This Happens:
  • A fundamental misunderstanding that the definition of degree strictly requires the differential equation to be a polynomial in all its derivatives (e.g., y', y'', y''').
  • Over-eagerness to apply the 'highest power of the highest order derivative' rule without first checking the prerequisite condition of polynomial form.
  • Confusion between functions of the independent variable (which are allowed) and functions of the derivatives of the dependent variable (which make the degree undefined). For example, `sin(x)` is fine, but `sin(dy/dx)` is not for degree definition.
✅ Correct Approach:

To correctly determine the order and degree of a differential equation:

  1. Identify the Order: This is the order of the highest derivative present in the equation. This is generally straightforward.
  2. Determine the Degree:
    • First, ensure the differential equation is free from radicals and fractional powers of all derivatives. Square/cube/raise to powers on both sides until all derivatives have integer powers.
    • Next, critically examine if the equation can be expressed as a polynomial in all its derivatives (`y'`, `y''`, `y'''`, etc.). This means no `sin(y')`, `e^(y'')`, `log(y''')`, etc.
    • If it IS a polynomial in derivatives, the degree is the highest power of the highest order derivative.
    • CRITICAL POINT: If it cannot be expressed as a polynomial in its derivatives (due to terms like `sin(y')`, `e^(y'')`, `log(y''')`, `tan(y')`), then the degree is undefined.
📝 Examples:
❌ Wrong:

For the differential equation: `d³y/dx³ + sin(dy/dx) = 0`

Many students might incorrectly state the degree as 1 (since `d³y/dx³` has power 1).

✅ Correct:

For the differential equation: `d³y/dx³ + sin(dy/dx) = 0`

  • Order: The highest derivative is `d³y/dx³`, so the order is 3.
  • Degree: The term `sin(dy/dx)` prevents the equation from being a polynomial in its derivatives. Therefore, the degree is undefined.
💡 Prevention Tips:
  • Always check the 'polynomial in derivatives' condition first before attempting to find the degree.
  • Be highly vigilant for terms involving derivatives inside transcendental functions (e.g., `sin(y')`, `cos(y'')`, `e^(y''')`, `log(y')`, `tan⁻¹(y'')`) or fractional powers of derivatives. These are strong indicators of an undefined degree.
  • For JEE Advanced, understanding this subtle but crucial point is essential, as 'undefined' is a common trap option.
JEE_Advanced
Critical Sign Error

❌ Misinterpretation of Negative Exponents or Reciprocal Derivative Terms when Determining Degree

Students often incorrectly determine the degree of a differential equation due to 'sign errors' in handling negative exponents or reciprocal derivative terms. This prevents correctly transforming the ODE into a polynomial in its derivatives, leading to an incorrect or undefined degree.
💭 Why This Happens:
  • Conceptual Gap: Misunderstanding that degree requires the ODE to be a polynomial in its derivatives.
  • Algebraic Mistakes: Errors clearing denominators or rationalizing negative exponents of derivative terms.
  • Rushing: Insufficient attention to algebraic manipulation.
✅ Correct Approach:

To correctly determine degree:

  1. Identify Order: Find the highest derivative.
  2. Rationalize: Eliminate all fractional and negative powers (reciprocals) of all derivatives.
  3. Polynomial Check: Ensure it's a polynomial in its derivatives. If not (e.g., sin(dy/dx)), degree is undefined.
  4. Determine Degree: The power of the highest order derivative in the rationalized, polynomial form.
📝 Examples:
❌ Wrong:

Consider the ODE: y'' + (dy/dx)-1 = x

Wrong Approach: Students might directly assume the degree is -1 (from (dy/dx)-1) or 1 (from y''). This is incorrect because the equation is not a polynomial in derivatives in this form due to the negative exponent.

✅ Correct:

For the ODE: y'' + (dy/dx)-1 = x

  1. The highest order derivative is y'', so Order = 2.
  2. To clear the negative exponent, multiply the entire equation by (dy/dx):
    y''(dy/dx) + 1 = x(dy/dx)
  3. Rearrange into polynomial form:
    y''(dy/dx) - x(dy/dx) + 1 = 0
  4. Now, the equation is a polynomial in its derivatives. The highest order derivative y'' has power 1.
  5. Therefore, the Degree = 1.
💡 Prevention Tips:
  • Always Rationalize: Before determining the degree, clear all fractional and negative powers of derivatives.
  • Verify Polynomial Form: Confirm the equation can be written as a polynomial in all its derivatives (e.g., no log(y') or ey'' terms, as their degree is undefined).
  • Careful Algebra: Be meticulous with algebraic manipulation steps to avoid sign-related errors.
  • JEE Advanced Specific: Expect complex transformations requiring careful algebraic handling.
JEE_Advanced
Critical Unit Conversion

❌ Ignoring Radical/Fractional Powers Before Determining Degree

A common and critical mistake when determining the degree of an Ordinary Differential Equation (ODE) is to directly identify the power of the highest order derivative without first ensuring the equation is a polynomial in its derivatives, i.e., free from radicals and fractional powers of derivatives. Students often overlook this crucial algebraic pre-processing step.
💭 Why This Happens:
This error primarily stems from a partial understanding of the definition of the degree of an ODE. While the order is simply the highest derivative present, the degree is defined as the power of the highest order derivative only after the equation has been rationalized and cleared of fractional/radical powers of all derivatives. Students frequently rush or forget this algebraic manipulation, especially under exam pressure.
✅ Correct Approach:
To correctly determine the degree of an ODE, follow these steps:
  1. First, identify the order of the ODE (the highest order derivative).
  2. Next, if the ODE contains any radical signs or fractional powers involving derivatives, algebraically manipulate the equation to eliminate them. This often involves squaring, cubing, or raising both sides to a suitable power.
  3. Once the equation is a polynomial in its derivatives (i.e., all derivatives have integer powers), identify the power of the highest order derivative found in step 1. This power is the degree of the ODE.
JEE Advanced Tip: Questions are often designed to test this specific understanding by presenting ODEs with radicals or fractional powers.
📝 Examples:
❌ Wrong:
Consider the ODE:
(d²y/dx²) = √[1 + (dy/dx)²]
Incorrect approach: The highest derivative is d²y/dx² (order 2). Its power is 1. So, the degree is 1.
Reason for error: The equation contains a radical involving a derivative, which must be eliminated first.
✅ Correct:
Using the same ODE: (d²y/dx²) = √[1 + (dy/dx)²]
Correct approach:
  1. Order: The highest order derivative is d²y/dx², so the order is 2.
  2. Rationalize: Square both sides to remove the radical:
    (d²y/dx²)² = 1 + (dy/dx)²
  3. Degree: Now, the highest order derivative is d²y/dx², and its power is 2. Therefore, the degree is 2.
💡 Prevention Tips:
  • Understand the Definition: Memorize and truly understand that degree requires the ODE to be a polynomial in its derivatives.
  • Always Check for Radicals/Fractions: Before stating the degree, make a conscious check for any square roots, cube roots, or fractional exponents on derivative terms.
  • Practice Algebraic Manipulation: Work through examples involving squaring, cubing, or raising to higher powers to clear radicals effectively.
  • CBSE vs. JEE: While CBSE might sometimes give simpler cases, JEE Advanced will often specifically include these 'traps' to test your thorough understanding.
JEE_Advanced
Critical Calculation

❌ Incorrectly Determining Degree in Presence of Radicals or Fractional Powers

A common critical mistake in JEE Advanced is determining the degree of a differential equation without first transforming it into a polynomial in its derivatives. Students often directly take the power of the highest order derivative, even if the equation contains radicals or fractional exponents involving derivatives, leading to an incorrect degree.
💭 Why This Happens:
This error stems from an incomplete understanding of the definition of the degree of a differential equation. The degree is defined only when the differential equation can be expressed as a polynomial in its derivatives. If radicals or fractional powers of derivatives are present, the equation is not yet in polynomial form, and direct identification of the degree is invalid.
✅ Correct Approach:
For an ordinary differential equation (ODE) to determine its degree, follow these steps:
  • First, identify the order of the differential equation, which is the order of the highest derivative present.
  • Next, eliminate all radicals and fractional powers from the derivatives by raising both sides of the equation to appropriate integral powers. The goal is to make the equation a polynomial in its derivatives.
  • Once the equation is free of radicals and fractional powers of derivatives, the degree is the power of the highest order derivative, provided this derivative appears in the equation after simplification.
📝 Examples:
❌ Wrong:
Consider the equation:
(d^2y/dx^2) = (1 + (dy/dx)^2)^(3/2)
Wrong approach: Student might identify the highest order derivative as d^2y/dx^2 (order = 2) and incorrectly state the degree as 3 (from the power 3/2, or just 3). This is wrong because the equation is not a polynomial in derivatives due to the fractional power 1/2.
✅ Correct:
Using the same equation:
(d^2y/dx^2) = (1 + (dy/dx)^2)^(3/2)
Correct approach:
  1. Isolate the radical/fractional power: Already done.
  2. Square both sides to eliminate the 1/2 power:
    (d^2y/dx^2)^2 = (1 + (dy/dx)^2)^3
  3. Now, the equation is a polynomial in its derivatives.
  4. Identify highest order derivative: d^2y/dx^2, so Order = 2.
  5. Identify its power: The power of d^2y/dx^2 is 2. So, Degree = 2.
💡 Prevention Tips:
  • Always Simplify First: Before determining the degree, ensure the ODE is free of radicals or fractional powers involving any derivative.
  • Polynomial Check: Mentally verify if the equation is a polynomial in y, y', y'', ... terms. If not, manipulate it.
  • JEE Advanced Specific: This type of question is a frequent trap in JEE Advanced, designed to test the fundamental understanding of definitions. Pay close attention to powers.
JEE_Advanced
Critical Calculation

❌ Ignoring Radicals or Non-Polynomial Forms in Derivatives for Degree Calculation

A critical mistake students make is directly stating the degree of an ODE without first ensuring that the equation is a polynomial in its derivatives. This oversight often occurs when radicals (like square roots) or fractional powers are present involving derivatives, leading to an incorrect numerical degree or an undefined degree where it should be defined.
💭 Why This Happens:
This error stems from a fundamental misunderstanding or oversight of the precise definition of 'degree'. The degree of a differential equation is defined only when it can be expressed as a polynomial in all its derivatives. Students frequently rush through the problem or overlook the necessary algebraic manipulation to clear fractional powers or radicals involving derivative terms.
✅ Correct Approach:

  1. First, correctly identify the order of the differential equation, which is the order of the highest derivative present in the equation.

  2. To find the degree, the differential equation must be made free from radicals and fractional powers involving any derivative. This often requires algebraic steps such as squaring both sides, cubing both sides, or raising both sides to an appropriate integer power to clear the non-integer exponents.

  3. Once the equation is successfully expressed as a polynomial in its derivatives, the degree is defined as the highest power of the highest order derivative.

  4. If the equation cannot be expressed as a polynomial in its derivatives (e.g., contains terms like sin(dy/dx), e^(d²y/dx²), log(d³y/dx³)), then its degree is undefined.

📝 Examples:
❌ Wrong:

Consider the ODE:
(dy/dx) + √(d²y/dx²) = 0
Incorrect Approach: Many students would identify the order as 2 and mistakenly state the degree as 1 (from dy/dx) or 1/2 (from the square root term), without proper simplification.

✅ Correct:

For the same ODE: (dy/dx) + √(d²y/dx²) = 0
Correct Approach:



  1. First, determine the highest order derivative: d²y/dx². So, the Order = 2.

  2. To find the degree, isolate the radical and square both sides to eliminate the fractional power:
    √(d²y/dx²) = -(dy/dx)
    Square both sides:
    (d²y/dx²) = (-(dy/dx))²
    (d²y/dx²) = (dy/dx)²

  3. Now, the equation is a polynomial in its derivatives. The highest order derivative is d²y/dx², and its power is 1. Therefore, the Degree = 1.

💡 Prevention Tips:

  • Fundamental Definition: Always recall that the degree is defined only for ODEs that are polynomials in their derivatives. This is a crucial concept for both CBSE and JEE.

  • Simplify Aggressively: Before concluding the degree, algebraically manipulate the ODE to remove all radicals, fractional powers, or inverse trigonometric/logarithmic functions that enclose derivative terms. This is a common trick in JEE Main problems.

  • JEE Watch-out: Questions on order and degree are frequent in JEE Main and are often designed to test this specific algebraic simplification skill. Don't fall for the trap of direct inspection.

  • Practice Algebraic Skills: Strong algebraic manipulation skills are key here. Practice converting equations with radicals into polynomial forms.

JEE_Main
Critical Unit Conversion

❌ Misapplying Unit Concepts to Order and Degree of ODEs

A common critical mistake is incorrectly associating physical units with the order or degree of an Ordinary Differential Equation (ODE). Students might mistakenly believe that since the variables in an ODE often represent physical quantities with units (e.g., time in seconds, displacement in meters), the mathematical properties of the equation, such as its order or degree, should also possess units. This fundamental error stems from a misunderstanding that order and degree are purely dimensionless mathematical classifications, not quantities measurable in physical units.
💭 Why This Happens:
This confusion typically arises when students are concurrently studying the mathematical definitions of ODE properties and their applications in physics or engineering problems, where variables inherently carry units. The distinction between the units of the variables *within* the equation and the dimensionless nature of the equation's *mathematical structure* (order and degree) becomes blurred. Lack of explicit emphasis on the dimensionless nature of these properties can also contribute to this misconception.
✅ Correct Approach:
Always remember that the order (defined as the highest order derivative present in the ODE) and the degree (defined as the highest power of the highest order derivative when the ODE is expressed as a polynomial in its derivatives) are dimensionless positive integers. They are abstract mathematical properties used to classify and analyze ODEs, independent of any physical context or units of the variables involved. For JEE Main, a clear understanding of these definitions is crucial.
📝 Examples:
❌ Wrong:
A student, analyzing the equation d2y/dt2 + 3(dy/dt) + 2y = 0, might incorrectly state:
  • The order is 2 seconds-2.
  • The degree is 1 (unitless, but with a lingering thought that it 'should' have units if order did).
✅ Correct:
For the ODE d2y/dt2 + 3(dy/dt) + 2y = 0:
  • The order is 2. (Dimensionless integer)
  • The degree is 1. (Dimensionless integer)
These values are pure numbers, reflecting the mathematical structure of the equation, and have no associated physical units.
💡 Prevention Tips:
  • Conceptual Reinforcement: Understand that order and degree are classification tools for ODEs, similar to how integers classify numbers, and do not carry units.
  • Strict Definition Adherence: Revisit and memorize the precise definitions of order and degree. Focus on 'highest derivative' and 'highest power of the highest derivative' as purely mathematical concepts.
  • JEE Focus: In JEE Main, questions on order and degree are direct applications of their definitions. Do not overthink by trying to introduce physical unit concepts where they don't apply.
JEE_Main
Critical Sign Error

❌ Mishandling Algebraic Signs During Radical/Fractional Power Elimination for Degree Determination

Students frequently make critical sign errors during the algebraic manipulations required to convert a non-polynomial differential equation (e.g., involving radicals or fractional powers of derivatives) into a polynomial form. This often occurs when isolating terms before squaring or cubing, or incorrectly handling negative signs when raising terms to an even power, leading to an incorrect derived equation and thus, a wrong degree or an erroneous conclusion that the degree is 'not defined' when it actually is.
💭 Why This Happens:
  • Weak Algebraic Foundation: Lack of proficiency in algebraic manipulation, especially with signs and powers.
  • Misconception of `(-A)^2`: Forgetting that (-A)2 = A2, leading to persistence of a negative sign where it should be positive.
  • Rushing Simplification: Hasty steps without proper attention to sign changes when moving terms across the equality sign.
  • Incomplete Understanding of Degree: Failure to recognize that the equation *must* be polynomial in its derivatives before degree can be determined, and this process involves careful algebraic steps.
✅ Correct Approach:
To correctly determine the degree of a differential equation that is not initially polynomial in its derivatives:
  1. Isolate Radical/Fractional Power Term: Completely isolate the derivative term with the radical or fractional power on one side of the equation. Ensure all signs are handled correctly during this isolation.
  2. Eliminate Radical/Fractional Power: Raise both sides of the equation to the smallest integer power that will eliminate the radical/fractional power. Crucially, remember that (-A)n = An if 'n' is even, and (-A)n = -An if 'n' is odd.
  3. Repeat if Necessary: If other radical/fractional power terms remain, repeat steps 1 and 2 until the equation is entirely free of such terms involving derivatives.
  4. Identify Order and Degree: Once the equation is a polynomial in its derivatives, the order is the highest order derivative present. The degree is the highest power of this highest order derivative.
📝 Examples:
❌ Wrong:
Consider the ODE: 
 (d2y/dx2)1/2 + dy/dx = 0
A common sign error during simplification leading to an incorrect degree:
1. Student correctly isolates the radical: 
 (d2y/dx2)1/2 = -dy/dx
2. Student then squares both sides, but makes a sign mistake, incorrectly writing: 
 d2y/dx2 = -(dy/dx)2
(Incorrectly assuming the negative sign persists outside the square, treating (-A)2 as -A2).
3. Based on this incorrect equation, they might still state Order = 2 and Degree = 1. While the order is correct, the degree is derived from an algebraically flawed equation.
✅ Correct:
For the same equation: 
 (d2y/dx2)1/2 + dy/dx = 0
The correct approach is:
1. Isolate the radical term: 
 (d2y/dx2)1/2 = -dy/dx
2. Square both sides carefully, remembering that (-A)2 = A2:
 ((d2y/dx2)1/2)2 = (-dy/dx)2
This correctly simplifies to: 
 d2y/dx2 = (dy/dx)2
3. Rearranging to form a polynomial in derivatives: 
 d2y/dx2 - (dy/dx)2 = 0
4. The highest order derivative is d2y/dx2. Therefore, the Order = 2.
5. The power of the highest order derivative (d2y/dx2) in this polynomial form is 1. Therefore, the Degree = 1.
(JEE Tip: The degree is the power of the highest order derivative *after* the equation has been made polynomial in all its derivatives, not necessarily the highest power overall if it belongs to a lower order derivative.)
💡 Prevention Tips:
  • Master Algebraic Rules: Dedicate time to thoroughly practice algebraic manipulations, especially involving exponents and signs.
  • Double-Check Squaring/Cubing: Always verify the outcome of raising terms to a power, particularly with negative signs (remember (-x)2 = x2).
  • Systematic Isolation: Before taking powers, ensure the radical/fractional power term is fully and correctly isolated on one side of the equation.
  • Verify Polynomial Form: After all manipulations, mentally (or physically) confirm that the equation is indeed a polynomial in *all* its derivatives before determining the degree.
  • CBSE vs. JEE: Both exams test these concepts similarly. Precision in algebraic steps is paramount for both.
JEE_Main
Critical Approximation

❌ Incorrectly Assigning Degree When the ODE is Not a Polynomial in Derivatives

Students often attempt to find the degree of a differential equation even when it cannot be expressed as a polynomial in its derivatives. This leads to an incorrect value for the degree, or a failure to recognize that the degree is undefined.
💭 Why This Happens:
This critical mistake arises from a fundamental misunderstanding of the definition of the 'degree' of a differential equation. The degree is strictly defined only when the differential equation can be written as a polynomial in its derivatives. If terms like sin(dy/dx), e^(d^2y/dx^2), or log(d^3y/dx^3) are present, the equation is not a polynomial in its derivatives, and thus, its degree is undefined. Students often overlook this condition in a rush or due to conceptual gaps.
✅ Correct Approach:
To correctly determine the order and degree:
📝 Examples:
❌ Wrong:
Consider the ODE: e^(d^2y/dx^2) + dy/dx + y = 0
Wrong Approach: A student might identify the highest order derivative as d^2y/dx^2, so the order is 2. Then, they might incorrectly state the degree as 1 (assuming e^X has a power of 1). This is incorrect.
✅ Correct:
Consider the ODE: e^(d^2y/dx^2) + dy/dx + y = 0
Correct Approach:
1. Identify the highest order derivative: d^2y/dx^2. So, the Order = 2.
2. Check if the equation is a polynomial in its derivatives. The term e^(d^2y/dx^2) cannot be expressed as a polynomial in d^2y/dx^2. Therefore, the Degree is undefined.

JEE Specific Callout: For JEE, questions often test this specific condition. If a differential equation contains transcendental functions (like trigonometric, exponential, or logarithmic functions) of derivatives, its degree is always undefined.
💡 Prevention Tips:
  • Understand Definitions: Clearly memorize that the degree is only defined for ODEs that are polynomials in their derivatives.
  • Check for Transcendental Functions: Always scan the ODE for terms like sin(y'), cos(y''), e^(y'''), log(y'). If present, the degree is undefined.
  • Simplify Before Deciding: If radical or fractional powers are present, clear them first to convert the equation into polynomial form (if possible). Only then determine the degree.
  • Practice Variety: Solve numerous problems, including those where the degree is undefined, to solidify this concept.
JEE_Main
Critical Other

❌ Incorrectly determining degree when ODE is not a polynomial in its derivatives

Students often overlook the crucial condition that a differential equation must be expressible as a polynomial in its derivatives for its degree to be defined. They might try to determine the degree even when transcendental functions (like sin, cos, e^x) involve derivatives, or when derivatives appear under radicals or fractional powers, without first making the equation rational and integral in terms of derivative powers.
💭 Why This Happens:
This mistake stems from a fundamental misunderstanding of the precise definition of 'degree'. Many students rush to identify the highest power of the highest order derivative without ensuring the differential equation is first simplified to be free from radicals, fractional powers, or transcendental functions of the derivatives. They incorrectly apply the definition to non-polynomial forms.
✅ Correct Approach:

  1. First, identify the order of the differential equation, which is the order of the highest derivative present. This step is independent of the polynomial nature.

  2. For degree, ensure the differential equation is a polynomial in its derivatives. This means all derivatives must appear with only integer powers and not be inside radicals, fractional powers, or transcendental functions (sin, cos, log, e^x, etc.).

  3. If radicals or fractional powers of derivatives are present, rationalize the equation by raising both sides to an appropriate power to eliminate them. This step is crucial.

  4. Once the equation is made free of radicals and fractional powers of derivatives (and is a polynomial in derivatives), the degree is the highest power of the highest order derivative.

  5. JEE Specific Critical Point: If the differential equation cannot be expressed as a polynomial in its derivatives (e.g., sin(dy/dx), e^(d²y/dx²), log(d³y/dx³)), then its degree is not defined. This is a very common trap in JEE Main.

📝 Examples:
❌ Wrong:

Consider the equation: sin(dy/dx) + x²y = 0


Incorrect approach: Students might mistakenly identify the order as 1 and attempt to assign a degree (e.g., 1, or undefined without proper reasoning). The degree is undefined here because sin(dy/dx) cannot be expanded into a polynomial of dy/dx.

✅ Correct:

Example 1: Rationalizing the equation


Consider: √(dy/dx) = d²y/dx² + y



  • Order: The highest derivative is d²y/dx², so the order is 2.

  • Degree: The equation is not a polynomial in its derivatives due to √(dy/dx). To make it a polynomial, square both sides:

    • (dy/dx) = (d²y/dx² + y)²


    (dy/dx) = (d²y/dx²)² + 2y(d²y/dx²) + y²


  • Now, the highest order derivative is d²y/dx², and its highest power is 2. So, the degree is 2.


Example 2: Degree Not Defined


Consider: e^(d³y/dx³) + x(d²y/dx²) = 0



  • Order: The highest derivative is d³y/dx³, so the order is 3.

  • Degree: The term e^(d³y/dx³) makes the equation non-polynomial in its derivatives. It cannot be simplified into a polynomial form. Therefore, the degree is not defined.

💡 Prevention Tips:

  • Master Definitions: Always recall the precise definitions of order and degree before attempting a problem. Understand the 'polynomial in derivatives' condition thoroughly.

  • Systematic Check: First determine order, then rigorously check if the equation is a polynomial in its derivatives before trying to find the degree.

  • Rationalize First: If fractional powers or radicals involving derivatives are present, always rationalize the equation by raising both sides to an appropriate power.

  • Beware of Transcendental Functions: Be extremely cautious when derivatives are arguments of trigonometric (sin, cos, tan), exponential (e^x), or logarithmic (log x) functions. In such cases, the degree is almost always undefined for JEE.

  • Practice JEE Problems: Solve previous year JEE problems that specifically test this concept, as they often include tricky variations that hinge on these definitions.

JEE_Main
Critical Conceptual

❌ <span style='color: #FF0000;'>Incorrectly Determining Degree for Non-Polynomial ODEs</span>

Students often incorrectly assign a degree to an Ordinary Differential Equation (ODE) without first ensuring it's a polynomial in its derivatives, free from fractional powers or radicals. For example, if terms like sin(dy/dx) or (d²y/dx²)^(1/2) are present, they might assign a degree instead of recognizing it as not defined. This is a critical conceptual error for CBSE 12th exams.
💭 Why This Happens:
This critical error stems from misunderstanding the fundamental definition of an ODE's degree. Students frequently rush to identify the highest power of the highest order derivative without first verifying the crucial 'polynomial in derivatives' condition or clearing fractional powers/radicals. This conceptual gap leads to significant marks deduction.
✅ Correct Approach:
  1. Order: First, identify the order of the differential equation (the order of the highest derivative present).
  2. Polynomial Check: Next, ensure the equation is a polynomial in all its derivatives. This means no derivative should be inside non-algebraic functions (e.g., sin, cos, log, exp).
  3. Clear Radicals: If derivatives have fractional powers or are under radicals, eliminate them by raising both sides of the equation to an appropriate integer power.
  4. Degree: If steps 2 & 3 are satisfied, the power of the highest order derivative is the degree. Otherwise, the degree is not defined.
📝 Examples:
❌ Wrong:
Consider the ODE: dy/dx + sin(d²y/dx²) = 0.
Wrong approach: Stating order = 2, and degree = 1. (Incorrectly assumes polynomial form in derivatives).
✅ Correct:
  1. For dy/dx + sin(d²y/dx²) = 0:
    • Order: 2 (due to d²y/dx²).
    • Degree: Not defined, because sin(d²y/dx²) prevents the equation from being a polynomial in its derivatives.
  2. For (d²y/dx²)^(3/2) = dy/dx + 5:
    • Order: 2.
    • Simplify: Square both sides to remove the fractional power: (d²y/dx²)^3 = (dy/dx + 5)².
    • Now, it's a polynomial. The highest order derivative is d²y/dx² with power 3. So, Degree = 3.
💡 Prevention Tips:
  • Core Definition: Always remember that the degree is only defined for ODEs that are polynomials in their derivatives, free from fractional powers or radicals.
  • Systematic Steps: Follow a consistent procedure: 1. Determine Order, 2. Check for polynomial form and clear radicals/fractions, 3. Then determine the Degree.
  • Practice Diverse Problems: Work through examples involving trigonometric, exponential, and radical forms of derivatives to solidify your understanding.
CBSE_12th

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Ordinary differential equations (ODEs): order and degree

Subject: Mathematics
Unit: 9 - DIFFERENTIAL EQUATIONS
Sub-unit: 9.1 - Basics
Complexity: High
Syllabus: JEE_Main

Content Completeness: 55.6%

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📚 Explanations: 0
📝 CBSE Problems: 18
🎯 JEE Problems: 18
🎥 Videos: 0
🖼️ Images: 0
📐 Formulas: 3
📚 References: 10
⚠️ Mistakes: 57
🤖 AI Explanation: No