Differential Equations Class 12 Notes

The Differential Equations Class 12 Notes cover Class 12 Mathematics Chapter 9 Differential Equations in 38 pages of concise theory, solved examples and end-of-section summaries. The Differential Equations Class 12 Notes explain order, degree, variable-separable, linear and homogeneous differential equations, with each result followed by the conditions under which it holds. The notes is downloadable as a free file aligned to the 2026-27 NCERT syllabus.

CBSE Boards: 5 to 7 marks (Calculus unit overall carries about 35 marks)
JEE Main: 1 to 2 questions on order, degree, linear (integrating factor) and homogeneous DEs (about 3 to 5% of Maths)
CUET (UG) Maths: 2 to 3 direct MCQs on order/degree, general/particular solution, and variable separable form every year

Chapter 9 Differential Equations Notes PDF

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Pages: 26 Methods: 3 standard Solved formulas: 14 Syllabus: 2026-27

If you want to walk into the boards with a clean drill on identifying the type of differential equation first, and only then applying the correct solution recipe, the Differential Equations Class 12 Notes below give you the definitions, the three solving methods, and solved templates in the exact NCERT order.

These Collegedunia the Differential Equations Class 12 Notes are compiled by Class 12 Maths specialists and cross-checked against the 2026-27 NCERT textbook and the past five years of CBSE marking schemes, so what you revise lines up with what examiners actually award marks for.

Differential Equations Notes - Class 12 Maths

Class 12 Maths Chapter 9 Differential Equations: Topic-by-Topic Breakdown

Differential Equations is a classification-then-method chapter. The single biggest jump in marks comes from correctly identifying whether a DE is variable-separable, homogeneous, or linear before you start solving. Below is the concept walkthrough in the exact order the NCERT chapter introduces it, with the formulas you must commit to memory.

1. What is a differential equation? Order and degree

A differential equation is an equation involving an unknown function and its derivatives. The order is the order of the highest-order derivative that appears, and the degree is the exponent of that highest-order derivative once the equation is made a polynomial in derivatives, free of radicals and fractions involving the derivatives.

For example, ( d² ydx² )³ + 2 ( dydx )⁵ + y = 0 has order 2 and degree 3. Degree is defined only when the equation can be expressed as a polynomial in derivatives; otherwise degree is not defined.

Quick Note: If the equation contains sin(dydx) or e^dy/dx , the degree is not defined, a recurring 1-mark MCQ trap.

2. General and particular solutions

A general solution of a differential equation of order n is a relation between the variables containing n arbitrary constants and which satisfies the equation. A particular solution is obtained from the general solution by giving particular values to those constants, usually using initial conditions like y(0) = 1 .

For instance, y = A cos x + B sin x is the general solution of d² ydx² + y = 0 (two arbitrary constants because the order is 2), and y = 2 cos x + 3 sin x is a particular solution corresponding to A = 2, B = 3 .

Common Confusion: Students forget that the number of arbitrary constants in the general solution always equals the order of the DE. A first-order DE has exactly one arbitrary constant; a second-order DE has two.

3. Formation of a differential equation from a given family of curves

If a family of curves contains n arbitrary constants, then the differential equation representing it is obtained by differentiating the relation n times and eliminating the n constants from the resulting equations. The DE that emerges has order n.

4. Variable-separable differential equations

5. Homogeneous differential equations

6. Linear differential equations of the first order

7. The four-step recipe for solving any Chapter 9 DE

Types of differential equations: ordinary, partial, linear, non-linear

Differential Equations Video Walkthrough

Source: Magnet Brains on YouTube

How will Collegedunia's NCERT Notes for Class 12 Maths Chapter 9 help you?

Order and degree definitions are given with edge cases (radicals, fractional powers, transcendental functions of derivatives) so the 1-mark MCQs become automatic.
The general-vs-particular solution distinction is anchored to the order of the DE (number of arbitrary constants), the way CBSE 1-mark and 2-mark questions test it.
Each solving method, variable separable, homogeneous y = vx substitution, and linear integrating factor, is given as a templated 4-step process matched to a board-style example.
The "is it linear in y or in x?" check is highlighted, since switching the dependent variable is the single most-missed step in the 5-mark question.

Class 12 Maths Chapter 9 Differential Equations: Most Important Topics

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

NCERT Class 12 Maths Chapter 9 Differential Equations Notes: Important Topics
Definition of a differential equation	Order and degree of a DE
General and particular solutions	Initial-value problems
Formation of a DE from a family of curves	Verifying a given function is a solution
Variable-separable differential equations	Standard integrals appearing in DE solutions
Homogeneous functions of degree zero	Substitution y = vx for homogeneous DEs
Linear DEs dydx + Py = Q	Integrating factor e^{∫ P dx}
Linear DEs in x : dxdy + P₁ x = Q₁	Applications to growth/decay and basic exponential models