Differential Equations Class 12 Notes - Class 12 Maths Ch 9

These Class 12 Maths Chapter 9 Differential Equations notes cover order, degree, and the three main solving methods. Each result comes with the conditions under which it holds, in the exact NCERT order. You can download the full notes as a free PDF right here on this page, aligned to the 2026-27 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

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

Differential Equations is a classification-then-method chapter. Most marks come from first spotting whether a DE is variable-separable, homogeneous, or linear. The concepts below follow the NCERT order.

1. What is a differential equation? Order and degree

A differential equation involves an unknown function and its derivatives. The order is the highest derivative present. The degree is the power of that highest derivative once the equation is a polynomial in derivatives.

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 an order-n DE has n arbitrary constants. A particular solution fixes those constants using initial conditions like y(0) = 1 .

Common Confusion: the number of arbitrary constants always equals the order. A first-order DE has one; a second-order DE has two.

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

If a family of curves has n arbitrary constants, differentiate n times and eliminate the constants. The resulting DE has order n.

4. The three solving methods

Variable-separable: write the DE as f(y) dy = g(x) dx, then integrate both sides.

Homogeneous: if dydx is a function of y/x, substitute y = vx to make it separable.

Linear first-order: for dydx + Py = Q, multiply by the integrating factor e^{∫ P dx}, then integrate.

Differential Equations Class 12 Solved Step by Step (Video)

Source: Magnet Brains on YouTube

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

• Order and degree definitions include the edge cases, so the 1-mark MCQs become automatic.
• Each solving method is given as a short recipe with a board-style example.
• The "linear in y or in x?" check is flagged, the most-missed step in the 5-mark question.

Class 12 Maths Chapter 9 Differential Equations: Most Important Topics

Differential Equations Class 12 Notes: Most Repeated Questions in Board Exams

Ques. Find the order and degree of (d³ ydx³)² + (d² ydx²)³ + dydx + 4 = 0 . (2024, 2022, 2019)

Ques. Solve dydx = x + yx . (2024, 2023, 2020)

Ques. Solve the linear DE dydx + 2y = e^3x . (2024, 2021, 2018)

Ques. Form the differential equation representing the family y = a sin(x + b) where a, b are arbitrary constants. (2023, 2020)

Differential Equations Notes: Previous Year Paper Analysis

• Every paper carries at least one 1-mark MCQ on order, degree, or the number of arbitrary constants in a general solution.
• Linear DEs (integrating-factor method) have appeared as a 5-mark question in 4 of the last 5 papers, treat it as a guaranteed question.
• Homogeneous DEs ( y = vx ) appear roughly every alternate year for 3 or 5 marks.

Weightage of Differential Equations in JEE Main, CUET, and CBSE 2026

Other Resources for Class 12 Maths Chapter 9 Differential Equations

• NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations
• Class 12 Maths Chapter 9 Differential Equations Formula Sheet
• Class 12 Maths Chapter 9 Differential Equations Handwritten Notes

Common Mistakes Students Make When Using the Differential Equations Class 12 Notes

• Confusing order with degree. Order is about which derivative is highest; degree is about the power of that highest derivative once the equation is polynomial in derivatives. Always state both separately.
• Forgetting that degree is not defined when the DE contains sin(dy/dx) , e^dy/dx , or log(dy/dx) . Writing "degree = 1" in such cases is automatic mark-loss.
• Skipping the integrating factor exponent. Writing IF = ∫ P dx instead of e^{∫ P dx} . The exponential is mandatory.
• Missing the linear-in-x form. An equation may not be linear in y but linear in x. Switch the dependent variable before you solve. Missing this switch is the most costly 5-mark error.

NCERT Notes for Class 12 Maths: All Chapters

Differential Equations Class 12 Notes - Quick Summary

• First identify the DE type, then apply the matching method.
• Order = highest derivative; degree = its power once the equation is a polynomial in derivatives.
• Three methods to master: variable-separable, homogeneous (y = vx), and linear (integrating factor).
• The integrating-factor method is a near-certain 5-mark board question.

Exercise-wise Breakdown of the Differential Equations Chapter

PDF Download Formats and Languages for the Differential Equations Chapter

• NCERT-faithful: every definition and exercise matches the printed textbook.
• Hindi edition: the same PDF is available in Hindi, with matching page numbers.

How the Differential Equations Notes Pair with NCERT Solutions and the Formula Sheet

How to Use the Differential Equations Notes Page Most Effectively