Class 12 Maths Chapter 9: Differential Equations

Differential Equations is Chapter 9 of NCERT Class 12 Maths. It explains what a differential equation is, how to find its order and degree, and how to solve three important first order types: variables separable, homogeneous and linear equations. Watch the five video lessons below, then use these notes, formulas and solved examples to revise.

What you will learn

The meaning of a differential equation, its order and its degree
The difference between a general solution and a particular solution
Solving equations by separating the variables
Solving homogeneous differential equations with the substitution y = vx
Solving linear differential equations using the integrating factor

Video Lessons

Differential Equations - Part 1

Differential Equations - Part 2

Differential Equations - Part 3

Differential Equations - Linear Differential Equations

Differential Equations - Order, Degree and Miscellaneous Questions

Key Concepts and Formulas

What Is a Differential Equation

An equation that involves the derivative of a dependent variable with respect to an independent variable is called a differential equation. In this chapter we deal only with ordinary differential equations, where there is a single independent variable.

Notation: y′ = dy/dx, y″ = d²y/dx²
Example:  dy/dx + 3y = x²

Order and Degree

The order tells you the highest derivative present, while the degree tells you the power of that highest derivative. Degree makes sense only when the equation is a polynomial in its derivatives.

Order = order of the highest derivative in the equation (always a positive integer)
Degree = highest power of the highest order derivative, when the equation is a polynomial in derivatives
If a derivative appears inside sin, cos, log, eˣ and so on, the degree is not defined

General and Particular Solutions

A solution is a function that satisfies the equation. The general solution contains arbitrary constants, and fixing them with given conditions produces a particular solution.

Family of solution curves y = x² + C with one particular solution, Class 12 Maths Differential Equations — Each C gives a curve; y(1) = 2 picks y = x² + 1

General solution: contains as many arbitrary constants as the order of the equation
Particular solution: obtained by finding the constants from given initial conditions
To verify a solution, differentiate it and substitute in the equation

Variables Separable Method

If dy/dx can be written as a product g(x) × h(y), bring all y terms to one side and all x terms to the other, then integrate both sides. Add the constant only once, on one side.

dy/dx = g(x) h(y)
⇒ ∫ dy/h(y) = ∫ g(x) dx + C,  h(y) ≠ 0

Homogeneous Differential Equations

A function F(x, y) is homogeneous of degree zero if F(λx, λy) = F(x, y). An equation dy/dx = F(x, y) with such a function is called homogeneous and becomes variables separable after a substitution. After integrating, replace v by y/x to get the answer in x and y.

Test: F(λx, λy) = λ⁰ F(x, y)
Put y = vx, so dy/dx = v + x dv/dx
For dx/dy = G(x, y): put x = vy

Linear Equations of First Order

A first order equation in which y and dy/dx appear only to the first power and are not multiplied together is linear. Here P and Q are functions of x only. The integrating factor turns the left side into the derivative of a product.

Standard form: dy/dx + P y = Q
I.F. = e^(∫P dx)
Solution: y × (I.F.) = ∫ Q × (I.F.) dx + C
Linear in x: dx/dy + P₁x = Q₁
I.F. = e^(∫P₁ dy)
x × (I.F.) = ∫ Q₁ × (I.F.) dy + C

How to choose between variables separable, homogeneous and linear methods, Class 12 Differential Equations — Match the form of the equation to the method

Solved Examples

Example 1: Find the order and degree of (d²y/dx²)³ + 4(dy/dx)⁵ − y = 0.

Highest derivative = d²y/dx², so order = 2
Its power = 3
∴ Order = 2, Degree = 3

Example 2: Find the order and degree of d²y/dx² + sin(dy/dx) = 0.

Highest derivative = d²y/dx², so order = 2
dy/dx appears inside sine: not a polynomial
∴ Order = 2, Degree not defined

Example 3: Show that y = A cos 3x + B sin 3x is a solution of y″ + 9y = 0.

y′ = −3A sin 3x + 3B cos 3x
y″ = −9A cos 3x − 9B sin 3x = −9y
y″ + 9y = −9y + 9y = 0
∴ y = A cos 3x + B sin 3x is a solution

Example 4: Solve dy/dx = 3x² e⁻ʸ given y = 0 when x = 0.

eʸ dy = 3x² dx
∫ eʸ dy = ∫ 3x² dx
eʸ = x³ + C
x = 0, y = 0: 1 = 0 + C, so C = 1
∴ eʸ = x³ + 1

Example 5: Solve the homogeneous equation dy/dx = (x + 2y)/x.

Put y = vx: v + x dv/dx = 1 + 2v
x dv/dx = 1 + v
dv/(1 + v) = dx/x
log|1 + v| = log|x| + log k
1 + y/x = kx
∴ y = kx² − x

Example 6: Solve dy/dx + 2y = 6eˣ.

P = 2, Q = 6eˣ
I.F. = e^(∫2 dx) = e²ˣ
y e²ˣ = ∫ 6eˣ × e²ˣ dx = ∫ 6e³ˣ dx
y e²ˣ = 2e³ˣ + C
∴ y = 2eˣ + C e⁻²ˣ

Example 7: Solve x dy/dx + y = x³ for x > 0.

Divide by x: dy/dx + y/x = x²
I.F. = e^(∫dx/x) = e^(log x) = x
xy = ∫ x² × x dx = x⁴/4 + C
∴ y = x³/4 + C/x

Common Mistakes to Avoid

Before finding the degree, check that the equation is a polynomial in derivatives; clear any radicals or fractional powers first.
In linear equations, bring the equation to standard form with coefficient of dy/dx equal to 1 before reading off P. Students often take P from the unreduced equation.
Remember that e^(log x) = x; simplify the integrating factor before multiplying.
When a condition like y(0) = 1 is given, the question wants a particular solution; stopping at the general solution loses marks.

Quick Revision

Separable: ∫ dy/h(y) = ∫ g(x) dx + C
Homogeneous: y = vx, dy/dx = v + x dv/dx
Linear: dy/dx + Py = Q
I.F. = e^(∫P dx)
y × I.F. = ∫ Q × I.F. dx + C

Order = highest derivative; degree = its power (only if the equation is a polynomial in derivatives)
Number of arbitrary constants in the general solution = order