Class 12 Maths Notes

Differential Equations Class 12 Notes

Complete, exam-ready notes on differential equations: order and degree, forming a differential equation from a family of curves, variable-separable and homogeneous equations, and the first-order linear equation solved with an integrating factor — written for CBSE boards and JEE revision.

Class12SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

A differential equation links a function to its derivatives, and solving it means recovering the function — separable, homogeneous and linear equations cover almost every exam case.

Order and Degree

Differential equation

An equation involving derivatives of an unknown function. The order is the highest derivative present; the degree is the power of the highest derivative after the equation is free of radicals and fractions.

Example:
$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} = \sin x$
has order 2.
To read the degree, first clear radicals in the highest derivative.
A general solution contains as many arbitrary constants as the order; a particular solution fixes them with initial conditions.

Order then degree

Compute the order first. Only after removing radicals/fractions touching the highest derivative can you state the degree correctly — squaring an equation can change its degree.

Forming a Differential Equation

Differentiate the general solution as many times as it has constants to eliminate them.
Example:
$y = A\cos x + B\sin x$
→
$y'' = -y$
(eliminating A and B).
The result is a differential equation of order equal to the number of constants eliminated.

y = A\cos x + B\sin x \Rightarrow \frac{d^2y}{dx^2} + y = 0

Eliminating constants

Variable-Separable Equations

Separable form

When the equation can be written $f(x)\,dx + g(y)\,dy = 0$ , integrate each side directly: the x-terms on one side, the y-terms on the other.

Write
$\frac{dy}{dx} = f(x)g(y)$
as
$\frac{dy}{g(y)} = f(x)\,dx$
.
Integrate both sides and add a single constant C.
Apply the initial condition last to find the particular solution.

\frac{dy}{g(y)} = f(x)\,dx \Rightarrow \int \frac{dy}{g(y)} = \int f(x)\,dx + C

Separation of variables

Homogeneous Equations

Homogeneous equation

An equation of the form $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$ . Substitute $y = vx$ , so $\frac{dy}{dx} = v + x\frac{dv}{dx}$ , to reach a separable form in v.

To test, put y = vx: every term in x and y becomes a power of x times a function of v.
After solving, substitute v = y/x back to express the answer in x and y.
Homogeneous equations frequently appear in JEE as short-answer questions.

y = vx,\quad \frac{dy}{dx} = v + x\frac{dv}{dx}

Homogeneous substitution

Linear Differential Equations

First-order linear

\frac{dy}{dx} + Py = Q

Where P and Q are functions of x only. Multiply by the integrating factor $\text{IF} = e^{\int P\,dx}$ so the left side becomes $\frac{d}{dx}(y\cdot\text{IF})$ .

Solution:
$y\cdot e^{\int P\,dx} = \int Q\,e^{\int P\,dx}\,dx + C$
.
Bring every equation into the standard 'dy/dx + Py = Q' form before applying the formula.
Common exams test e^x, sin x or xᵏ as Q with an easily integrated P.

y\cdot e^{\int P\,dx} = \int Q\,e^{\int P\,dx}\,dx + C

Solution of a linear differential equation

Solved Examples

Example: Solve $\frac{dy}{dx} = xy$ with $y(0) = 1$ .

Solution: Separating variables: $\frac{dy}{y} = x\,dx$ . Integrating: $\ln y = \frac{x^2}{2} + C$ , so $y = Ae^{x^2/2}$ . Using $y(0)=1$ gives $A = 1$ , hence $y = e^{x^2/2}$ .

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Separation of variables

\int \frac{dy}{g(y)} = \int f(x)\,dx + C

Homogeneous substitution

y = vx,\quad \frac{dy}{dx} = v + x\frac{dv}{dx}

Integrating factor

\text{IF} = e^{\int P\,dx}

Linear solution

y\cdot\text{IF} = \int Q\cdot\text{IF}\,dx + C

Standard form

\frac{dy}{dx} + Py = Q

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

Order = highest derivative; degree = power of the highest derivative, found after clearing radicals.
General solution has as many constants as the order.
Separable if dy/dx = f(x)g(y) — split and integrate both sides.
Homogeneous: substitute y = vx.
Linear: bring to dy/dx + Py = Q, multiply by IF = e^{∫P dx}.
Apply the initial condition only at the end.

FAQ

Common questions

What is the order and degree of a differential equation?

The order is the highest derivative in the equation. The degree is the power to which that highest derivative is raised, after the equation is cleared of radicals and fractions.

When is an equation variable-separable?

When it can be rearranged as f(x)dx + g(y)dy = 0 — x-terms on one side and y-terms on the other — so each side integrates separately.

What is the integrating factor?

For dy/dx + Py = Q, the factor IF = e^{∫P dx} makes the left side the derivative of y·IF, so the solution is y·IF = ∫Q·IF dx + C.

How is a homogeneous equation recognised?

It can be written as dy/dx = f(y/x). Substituting y = vx turns it into a separable equation in v, which you solve and then substitute back.

Continue learning

Mastering this chapter with live help

Notes help, but doubts clear fastest in a live class. Narayan Gurukul Academy (ClassApna) runs small-batch CBSE, JEE and NEET coaching from our Mohali centre and online — with daily doubt support and mock tests.

Book a Free Demo See plans & pricing

One-on-one guidance available · Live online classes across India