CLASS XII MATHEMATICS CHAPTER 9

Differential Equations

Complete Class XII Mathematics chapter notes covering differential equations, order and degree, formation, general and particular solutions, variable separable equations, homogeneous equations, linear differential equations and applications.

CBSE Pattern Order & Degree Formation Variable Separable Homogeneous Linear Differential Equations

Start Learning Enroll Now

Chapter Overview

Introduction

A differential equation is an equation involving one or more derivatives of a dependent variable with respect to an independent variable.

Differential equations are used to describe quantities that change continuously. They have applications in growth and decay, motion, population models and many physical situations.

dy/dx

Rate of Change

Represents the instantaneous rate of change of y with respect to x.

∫

Integration

Helps obtain solutions of many differential equations.

Constant

General solutions normally contain arbitrary constants.

Foundation

Basic Concepts

Differential Equation

An equation containing derivatives of a dependent variable with respect to an independent variable is called a differential equation.

dy/dx = f(x)

Ordinary Differential Equation

If a differential equation contains derivatives with respect to only one independent variable, it is called an ordinary differential equation.

dy/dx + y = x

First-order differential equation.

d²y/dx² + 4y = 0

Second-order differential equation.

Important Concept

Order and Degree

Order

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

3 d³y/dx³ + 2 d²y/dx² − dy/dx + y = 0

Order = 3

Degree

When a differential equation is polynomial in its derivatives, the degree is the highest power of the highest-order derivative after the equation has been made free from radicals and fractions involving derivatives.

(d²y/dx²)³ + 2(dy/dx)² + y = 0

Order = 2 Degree = 3

Formation

Formation of Differential Equations

A differential equation can be formed by eliminating arbitrary constants from a relation containing the dependent and independent variables.

Basic Procedure

Differentiate the given equation as many times as required.
Obtain equations involving the arbitrary constants.
Eliminate the arbitrary constants.
The resulting relation is the required differential equation.

Example

y = Ae^x + Be^−x

y = Ae^x + Be^−x

Differentiating:
y′ = Ae^x − Be^−x

Again differentiating:
y″ = Ae^x + Be^−x

Therefore,
y″ − y = 0

Solution Concepts

General and Particular Solution

General Solution

A solution containing arbitrary constants is called the general solution.

y = x² + C

Particular Solution

A solution obtained after assigning specific values to arbitrary constants is called a particular solution.

y = x² + 4

Initial Condition

An initial condition such as y(0)=2 helps determine the arbitrary constant and hence the particular solution.

Integration Method

Variable Separable Differential Equations

If a differential equation can be written so that all terms involving y occur on one side and all terms involving x occur on the other side, the variables are said to be separable.

dy/dx = f(x)g(y)

dy/g(y) = f(x) dx

Standard Method

Separate variables
Integrate both sides
Introduce the arbitrary constant C
Apply initial condition, if given

Example

dy/dx = xy

dy/y = x dx

Integrating:
∫ dy/y = ∫x dx

ln|y| = x²/2 + C

Therefore,
y = Ce^x²/2

Important Method

Homogeneous Differential Equations

A first-order differential equation is homogeneous if it can be expressed in the form

dy/dx = F(y/x)

Substitution

Put
y = vx

Then
dy/dx = v + x dv/dx

Example

dy/dx = 1 + y/x

Put
y = vx

Therefore,
dy/dx = v + x dv/dx

Hence,
v + x dv/dx = 1 + v

x dv/dx = 1

dv = dx/x

v = ln|x| + C

Since y = vx,
y = x(ln|x| + C)

Powerful Technique

First-Order Linear Differential Equation

The standard form of a first-order linear differential equation is:

dy/dx + P(x)y = Q(x)

Integrating Factor

IF = e^∫P(x)dx

y(IF) = ∫Q(x)(IF)dx + C

Example

dy/dx + y = e^x

Comparing with
dy/dx + Py = Q

P = 1
Q = e^x

Therefore,
IF = e^∫1dx = e^x

Hence,
y e^x = ∫e^x · e^x dx + C

y e^x = e^2x/2 + C

Therefore,
y = e^x/2 + Ce^−x

Real-World Applications

Applications of Differential Equations

📈

Growth

dy/dt = ky

Used in models where the rate of change is proportional to the quantity present.

📉

Decay

dy/dt = −ky

Used for quantities that decrease at a rate proportional to their amount.

🚗

Motion

v = ds/dt

Differential equations can model displacement, velocity and acceleration.

Exponential Growth Model

dy/dt = ky

y = Ce^kt

Geometrical Interpretation

Geometrical Meaning

For a curve represented by

dy/dx = f(x,y)

the value of dy/dx at a point gives the slope of the tangent to the solution curve at that point.

Worked Problems

Solved Examples

Example 1

Find the order and degree of:

(d³y/dx³)² + 5(d²y/dx²)³ + y = 0

Highest derivative:
d³y/dx³

Therefore,
Order = 3
The highest power of the highest-order derivative is 2.
Degree = 2

Example 2

Solve:

dy/dx = 2xy

Separate variables:

dy/y = 2x dx

Integrating:
ln|y| = x² + C

Hence,
y = Ce^x²

Example 3

Solve the linear differential equation:

dy/dx + 2y = e^x

Here,
P = 2, Q = e^x

Integrating factor:
IF = e^∫2dx = e^2x

Therefore,
y e^2x = ∫e^xe^2xdx + C

y e^2x = e^3x/3 + C

Hence,
y = e^x/3 + Ce^−2x

Must Remember

Formula Sheet

Order = highest order derivative

Degree = highest power of highest derivative

dy/dx = f(x)g(y)

dy/g(y) = f(x)dx

y = vx

dy/dx = v + x dv/dx

dy/dx + Py = Q

IF = e^∫Pdx

y(IF) = ∫Q(IF)dx + C

dy/dt = ky ⇒ y = Ce^kt

dy/dt = −ky ⇒ y = Ce^−kt

v = ds/dt

a = dv/dt

a = v dv/ds

Golden Rule

Before solving, first identify the type of differential equation. The method of solution depends on its form.

Practice Zone

Practice Questions

Q1. Find the order and degree of (d²y/dx²)⁴ + 3(dy/dx)² + y = 0.

Q2. Form the differential equation representing y = Ae^2x + Be^−2x.

Q3. Solve: dy/dx = x(1+y).

Q4. Solve the homogeneous differential equation dy/dx = (x+y)/(x−y).

Q5. Solve: dy/dx + 3y = 6e^x.

Q6. A quantity increases at a rate proportional to the quantity present. Form and solve the differential equation.

Higher Order Thinking

HOT Questions

HOT 1

A student claims that every first-order differential equation can be solved by separating variables. Examine the claim and give a suitable counterexample.

HOT 2

The differential equation dy/dx = (x+y)/(x−y) appears complicated. Explain why the substitution y=vx is appropriate and determine the resulting differential equation in v and x.

HOT 3

A family of curves is given by y = Ce^2x. Form the differential equation and explain why the arbitrary constant disappears.

HOT 4

Consider the equation dy/dx + P(x)y = Q(x). Explain why multiplying the equation by the integrating factor transforms its left-hand side into the derivative of a product.

HOT 5

A solution of a differential equation satisfies an initial condition. Explain how the initial condition changes a general family of solution curves into a unique particular solution.

Last-Minute Revision

Quick Revision

1. Identify the Type

First determine whether the equation is separable, homogeneous or linear.

2. Find Order

Look for the highest-order derivative.

3. Find Degree

Only after the equation is polynomial in its derivatives.

4. Apply Substitution

For homogeneous equations use y=vx.

5. Use Integrating Factor

For linear equations use IF=e^∫Pdx.

6. Apply Initial Condition

Use the given condition to determine C.

Differential Equation → Identify Type → Choose Method → Integrate → Apply Condition → Final Solution

Exam Strategy

Exam Tips

✓

Always identify the type of differential equation before starting calculations.

✓

In homogeneous equations remember y=vx and dy/dx=v+x dv/dx.

✓

In a linear differential equation compare carefully with dy/dx+Py=Q.

✓

Write the integrating factor explicitly before multiplying the equation.

✓

Do not forget the arbitrary constant C after integration.

✓

Check the final answer by differentiating it when time permits.

Build Strong Concepts

Master Differential Equations

Strengthen your concepts with structured chapter notes, practice questions, revision material and expert-guided mathematics coaching.

Enroll Now

← Back to Class XII Mathematics

← Chapter 8 Notes Chapter 9 Practice Test → Chapter 10 Notes →