Order and Degree of Differential Equations

Complete guide to understanding order and degree of differential equations for JEE

7 min read

Order and Degree of Differential Equations

Introduction

A Differential Equation (DE) is an equation involving derivatives of a function. Understanding the order and degree of a DE is fundamental for classification and solving techniques.

Differential Equation Definition

A differential equation is an equation containing derivatives of one or more dependent variables with respect to one or more independent variables.

Examples:

  1. $\frac{dy}{dx} + y = x$ (First-order DE)
  2. $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 3y = 0$ (Second-order DE)
  3. $\left(\frac{dy}{dx}\right)^3 + x\frac{dy}{dx} - y = 0$ (First-order, third-degree DE)

Order of Differential Equation

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

$$\boxed{\text{Order} = \text{Highest order derivative in the DE}}$$

Examples

Differential EquationOrderReason
$\frac{dy}{dx} = x + y$1Highest derivative: $\frac{dy}{dx}$
$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} = 0$2Highest derivative: $\frac{d^2y}{dx^2}$
$\frac{d^3y}{dx^3} + x^2\frac{dy}{dx} = \sin x$3Highest derivative: $\frac{d^3y}{dx^3}$
$y'' + 5y' + 6y = 0$2$y''$ is second derivative
$\left(\frac{d^2y}{dx^2}\right)^4 + \frac{dy}{dx} = 0$2Highest derivative: $\frac{d^2y}{dx^2}$ (power doesn’t matter)

Degree of Differential Equation

Degree: The degree of a differential equation is the power (exponent) of the highest order derivative, when the equation is a polynomial in derivatives.

$$\boxed{\text{Degree} = \text{Power of highest order derivative}}$$

Important Conditions

The degree is defined only when:

  1. The DE is a polynomial in all derivatives
  2. All derivatives are free from radicals and fractions

If the DE contains derivatives in radicals, exponents, or transcendental functions, degree is not defined.

Examples

Differential EquationOrderDegreeExplanation
$\frac{dy}{dx} + y = x$11Power of $\frac{dy}{dx}$ is 1
$\left(\frac{dy}{dx}\right)^3 + x\frac{dy}{dx} = y$13Power of $\frac{dy}{dx}$ is 3
$\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 6y = 0$21Power of $\frac{d^2y}{dx^2}$ is 1
$\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0$22Power of $\frac{d^2y}{dx^2}$ is 2
$\sqrt{\frac{dy}{dx}} = x + y$12After squaring: $\frac{dy}{dx} = (x+y)^2$
$\frac{d^2y}{dx^2} + \sqrt{\frac{dy}{dx}} = 0$2Not definedContains square root of derivative
$\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0$2Not definedContains transcendental function
$e^{dy/dx} = x$1Not definedDerivative in exponent

Steps to Find Order and Degree

Step 1: Identify the Highest Derivative

Look for the highest order derivative in the equation.

  • Order = Order of this derivative

Step 2: Check if DE is Polynomial in Derivatives

  • If yes, proceed to Step 3
  • If no, degree is not defined

Step 3: Make Derivatives Free from Radicals/Fractions

  • Remove square roots by squaring
  • Clear fractions if needed
  • Ensure all derivatives appear as positive integer powers

Step 4: Find Power of Highest Order Derivative

  • Degree = Exponent of highest order derivative

Interactive Demo: Slope Field Visualization

Explore how differential equations create direction fields. Click anywhere to draw solution curves that follow the slope at each point using Euler’s method.

Memory Tricks

🎯 Order Memory

“Order = Highest Derivative Number”

  • First derivative ($y'$) → Order 1
  • Second derivative ($y''$) → Order 2
  • Third derivative ($y'''$) → Order 3

Mnemonic: “Order comes from Order of derivative”

🎯 Degree Memory

“Degree = Power of Top Derivative”

  • Power of highest order derivative = Degree
  • Must be polynomial first!

Mnemonic: “Degree is the Derivative’s power”

🎯 When Degree Doesn’t Exist

“No REFT allowed”:

  • Radicals (square roots, cube roots)
  • Exponents (derivative in power)
  • Fractions (derivative in denominator)
  • Transcendental (sin, cos, log, etc.)

Common Mistakes to Avoid

❌ Mistake 1: Confusing Order with Degree

Wrong: In $\left(\frac{d^2y}{dx^2}\right)^3 = 0$, order = 3 ✗

Correct: Order = 2 (highest derivative is second), Degree = 3 (power of $\frac{d^2y}{dx^2}$) ✓

❌ Mistake 2: Finding Degree When Not Polynomial

Wrong: In $e^{dy/dx} + y = 0$, degree = 1 ✗

Correct: Degree is not defined (derivative appears in exponent) ✓

❌ Mistake 3: Forgetting to Simplify Before Finding Degree

Wrong: In $\sqrt{\frac{d^2y}{dx^2}} + y = 0$, degree not defined ✗

Correct: Square both sides: $\frac{d^2y}{dx^2} + 2y\sqrt{\frac{d^2y}{dx^2}} + y^2 = 0$ Still contains radical, so degree not defined ✓

Actually, this example shows degree is truly not defined!

❌ Mistake 4: Considering Lower Order Derivatives for Degree

Wrong: In $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^5 = 0$, degree = 5 ✗

Correct: Degree = 1 (power of highest order derivative $\frac{d^2y}{dx^2}$) ✓

Solved Examples

Example 1: Basic Order and Degree (JEE Main)

Find the order and degree of: $\frac{d^3y}{dx^3} + 2\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0$

Solution:

  • Highest derivative: $\frac{d^3y}{dx^3}$ → Order = 3
  • Polynomial in derivatives: Yes ✓
  • Power of $\frac{d^3y}{dx^3}$: 1 → Degree = 1

Answer: Order = 3, Degree = 1

Example 2: Degree After Simplification (JEE Main)

Find order and degree of: $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = x$

Solution: Squaring both sides:

$$1 + \left(\frac{dy}{dx}\right)^2 = x^2$$
  • Order: 1 (highest derivative is $\frac{dy}{dx}$)
  • Degree: 2 (power of $\frac{dy}{dx}$ after squaring)

Answer: Order = 1, Degree = 2

Example 3: Degree Not Defined (JEE Advanced)

Find order and degree of: $\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0$

Solution:

  • Order: 2 (highest derivative is $\frac{d^2y}{dx^2}$)
  • Degree: Not defined (contains $\sin$ of derivative, which is transcendental)

Answer: Order = 2, Degree = Not defined

Example 4: Complex Expression (JEE Advanced)

Find order and degree of: $\left[\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right]^{3/2} = 5\frac{d^3y}{dx^3}$

Solution: Raise both sides to power 2 to remove fractional power:

$$\left[\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right]^3 = 25\left(\frac{d^3y}{dx^3}\right)^2$$
  • Order: 3 (highest derivative is $\frac{d^3y}{dx^3}$)
  • Polynomial in derivatives: Yes (after simplification) ✓
  • Power of $\frac{d^3y}{dx^3}$: 2 → Degree = 2

Answer: Order = 3, Degree = 2

Example 5: Tricky Case (JEE Advanced)

Find order and degree of: $\frac{dy}{dx} + \sqrt{\frac{d^2y}{dx^2}} = 0$

Solution: Try to remove radical:

$$\frac{dy}{dx} = -\sqrt{\frac{d^2y}{dx^2}}$$

Square both sides:

$$\left(\frac{dy}{dx}\right)^2 = \frac{d^2y}{dx^2}$$
  • Order: 2
  • Degree: 1 (power of $\frac{d^2y}{dx^2}$ is 1)

Answer: Order = 2, Degree = 1

Practice Problems

Level 1: JEE Main Basics

Problem 1.1: Find order and degree of $\frac{dy}{dx} = x^2 + y^2$.

Solution
  • Order = 1 (highest derivative: $\frac{dy}{dx}$)
  • Degree = 1 (power of $\frac{dy}{dx}$ is 1)

Problem 1.2: Find order and degree of $\frac{d^4y}{dx^4} + 3y = 0$.

Solution
  • Order = 4 (highest derivative: $\frac{d^4y}{dx^4}$)
  • Degree = 1 (power of $\frac{d^4y}{dx^4}$ is 1)

Problem 1.3: Find order and degree of $\left(\frac{dy}{dx}\right)^4 + y^3 = x^5$.

Solution
  • Order = 1 (highest derivative: $\frac{dy}{dx}$)
  • Degree = 4 (power of $\frac{dy}{dx}$ is 4)

Level 2: JEE Main Advanced

Problem 2.1: Find order and degree of $y'' + (y')^3 + y = 0$.

Solution
  • Order = 2 (highest derivative: $y''$)
  • Degree = 1 (power of $y''$ is 1; $(y')^3$ doesn’t matter for degree)

Problem 2.2: Find order and degree of $\sqrt{\frac{d^3y}{dx^3}} + \frac{d^2y}{dx^2} = 0$.

Solution

Square both sides:

$$\frac{d^3y}{dx^3} + 2\frac{d^2y}{dx^2}\sqrt{\frac{d^3y}{dx^3}} + \left(\frac{d^2y}{dx^2}\right)^2 = 0$$

Still contains radical!

Order = 3, Degree = Not defined

Problem 2.3: Find order and degree of $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/4} = \frac{d^2y}{dx^2}$.

Solution

Raise to power 4:

$$\left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 = \left(\frac{d^2y}{dx^2}\right)^4$$
  • Order = 2
  • Degree = 4

Level 3: JEE Advanced

Problem 3.1: Find order and degree of $x\left(\frac{d^2y}{dx^2}\right)^3 + y\left(\frac{dy}{dx}\right)^4 + x^3 = 0$.

Solution

Already polynomial in derivatives.

  • Order = 2 (highest derivative: $\frac{d^2y}{dx^2}$)
  • Degree = 3 (power of $\frac{d^2y}{dx^2}$ is 3)

Problem 3.2: Find order and degree of $\frac{d^2y}{dx^2} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2}$.

Solution

Square both sides:

$$\left(\frac{d^2y}{dx^2}\right)^2 = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^3$$
  • Order = 2
  • Degree = 2

Problem 3.3: Determine if degree exists for $\frac{dy}{dx} = e^{d^2y/dx^2}$.

Solution

Taking logarithm:

$$\ln\left(\frac{dy}{dx}\right) = \frac{d^2y}{dx^2}$$

Derivative appears in logarithm (transcendental function).

Order = 2, Degree = Not defined

Classification of Differential Equations

By Order

  • First Order: $\frac{dy}{dx} = f(x, y)$
  • Second Order: $\frac{d^2y}{dx^2} = f\left(x, y, \frac{dy}{dx}\right)$
  • $n$-th Order: Highest derivative is $n$-th order

By Degree

  • First Degree: Linear in all derivatives
  • Higher Degree: Power of highest derivative > 1

By Linearity

  • Linear: No products/powers of $y$ and derivatives

    • Example: $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = x$

  • Non-linear: Contains products or powers

    • Example: $\frac{d^2y}{dx^2} + y\frac{dy}{dx} = 0$

Standard Forms

First Order, First Degree

$$\boxed{\frac{dy}{dx} = f(x, y)}$$

Examples:

  • $\frac{dy}{dx} = x + y$
  • $\frac{dy}{dx} = \frac{y}{x}$

Second Order, First Degree

$$\boxed{\frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = R(x)}$$

Example: $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 3y = e^x$

Quick Identification Table

FeatureHow to Identify
OrderCount apostrophes in $y', y'', y'''$ or look at denominator in $\frac{d^ny}{dx^n}$
DegreePower of highest derivative (if polynomial)
LinearNo $y \times y'$, no $(y')^2$, etc.
HomogeneousAll terms have same degree in $y$ and derivatives

Cross-References

Quick Revision Checklist

  • Order = highest derivative number
  • Degree = power of highest derivative (if polynomial)
  • Degree not defined for radicals/transcendental functions
  • Simplify to polynomial form before finding degree
  • Don’t confuse order with degree
  • Linear DE has no products/powers of $y$ and derivatives

Last updated: November 5, 2025