CBSE · Class 12 · Mathematics · Chapter 9

Differential
Equations

Complete chapter resources for CBSE Class 12 Maths — topic breakdown, key formulas, order & degree, solution methods, sample questions, previous year board questions, and instant AI question paper generation.

4Topics
6–8Board marks
8Sample questions
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Key Formulas — Chapter 9
  • Variable separable: f(x) dx = g(y) dy → ∫g(y)dy = ∫f(x)dx + C
  • Homogeneous DE: dy/dx = F(y/x); put y = vx
  • Linear DE (standard): dy/dx + P(x)y = Q(x)
  • Integrating factor: IF = e^(∫P dx)
  • General solution: y · IF = ∫(Q · IF) dx + C
  • Order: order of highest derivative present
Chapter Overview

What this chapter covers

A differential equation is an equation that involves an unknown function and one or more of its derivatives. Chapter 9 of NCERT Class 12 Mathematics begins by defining two fundamental attributes of every differential equation: its order (the order of the highest derivative present) and its degree (the power of that highest derivative, when the equation is written as a polynomial in its derivatives). Students also learn how to form a differential equation from a given family of curves by eliminating arbitrary constants through successive differentiation.

The chapter then covers three systematic solution methods. The variable separable method applies when all x-terms and dx can be moved to one side and all y-terms and dy to the other, allowing direct integration. The homogeneous method applies when the DE is of the form dy/dx = F(y/x); substituting y = vx converts it to a separable form. The linear differential equation method handles equations of the form dy/dx + P(x)y = Q(x) by computing an integrating factor IF = e^(∫P dx) and then integrating the resulting exact derivative.

Board questions in this chapter consistently test the ability to identify the correct solution method, set up the working cleanly, and add the constant of integration C. Word problems requiring a particular solution — where an initial condition (e.g., y = 1 when x = 0) is used to determine C — are high-probability long-answer questions carrying 5 marks and have appeared in recent CBSE Class 12 papers.

Topics

What's inside Chapter 9

As per NCERT Class 12 Mathematics (CBSE syllabus)

Topic 1
Order, Degree & Formation of Differential Equations
Defining order and degree of a DE. Identifying when degree is not defined (non-polynomial DEs). Forming a DE from a family of curves by eliminating arbitrary constants through differentiation.
Topic 2
Variable Separable Method
Rewriting dy/dx = f(x)·g(y) as dy/g(y) = f(x)dx, then integrating both sides. General solution involves one arbitrary constant C. Used when variables can be cleanly separated.
Topic 3
Homogeneous Differential Equations
Identifying homogeneous DEs where dy/dx = F(y/x). Applying the substitution y = vx (so dy/dx = v + x·dv/dx) to reduce to a separable form, then back-substituting v = y/x.
Topic 4
Linear Differential Equations
Standard form dy/dx + P(x)y = Q(x). Computing IF = e^(∫P dx). Multiplying both sides by IF to obtain d/dx[y·IF] = Q·IF, then integrating. Also handles dx/dy + P(y)x = Q(y) form.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Ch 7 · Integrals
All three solution methods require integration — standard forms, substitution, and partial fractions
Ch 5 · Continuity & Differentiability
Derivative rules and the chain rule used in forming and solving DEs
Chapter 9 Differential
Equations
Leads to
Ch 10 · Vector Algebra
Vectors are used in formulating DEs in three-dimensional physical problems
Engineering / JEE / Higher Mathematics
Second-order DEs, Laplace transforms, and modelling in physics & engineering
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 12 board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Objective 1 1–2 Order and degree identification, or nature of solution (general vs. particular)
Very Short Answer 2 1 Form the differential equation from a given family of curves; state the order
Short Answer 3 1 Solve a variable separable or homogeneous DE and find the general solution
Long Answer 5 1 Solve a linear DE and find the particular solution using an initial condition
Total (approximate) 6–8 3–4 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 12 Maths Chapter 9. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
The order and degree of the differential equation (d²y/dx²)³ + (dy/dx)² = sin x are: (a) order 2, degree 3 (b) order 2, degree 1 (c) order 1, degree 2 (d) order 3, degree 2
Q2 Easy 2 marks Short Answer
Form the differential equation representing the family of curves y = A cos x + B sin x, where A and B are arbitrary constants.
Q3 Medium 3 marks Short Answer
Solve the differential equation: dy/dx = (1 + y²)/(1 + x²), given that y = 1 when x = 0.
Q4 Medium 3 marks Short Answer
Show that the differential equation (x² + xy) dy = (x² + y²) dx is homogeneous. Hence solve it.
Q5 Medium 3 marks Short Answer
Find the general solution of the differential equation: x dy/dx + 2y = x² log x.
Q6 Hard 5 marks Long Answer
Solve the differential equation: (1 + x²) dy/dx + 2xy = 4x². Find the particular solution given that y = 0 when x = 0.
Q7 Hard 5 marks Word Problem
In a bank, principal P increases at a rate equal to 5% per year of the principal. If the initial principal is ₹1,000, form a differential equation and find the value of the principal after 10 years. (Use e^0.5 ≈ 1.6487)
Q8 Hard 5 marks Word Problem
A population grows at a rate proportional to the population present at time t. If the population doubles in 30 years, in how many years will the population triple? (Express answer in terms of log values.) Set up the differential equation, solve it, and find the required time.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 12 Maths board papers — Differential Equations chapter.

Board 20222 marks
Find the general solution of the differential equation: e^x tan y dx + (1 − e^x) sec² y dy = 0. (CBSE All India 2022)
Board 20235 marks
Solve the differential equation: x dy − y dx = √(x² + y²) dx, given that y = 0 when x = 1. (CBSE 2023, Set 1)
Board 20205 marks
Find the particular solution of the differential equation dy/dx + y cot x = 2x + x² cot x (x ≠ 0), given that y = 0 when x = π/2. (CBSE Delhi 2020)
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FAQ

Questions teachers ask

How many marks does Differential Equations carry in the CBSE Class 12 board exam? +
Differential Equations typically carries 6–8 marks in the CBSE Class 12 Maths board exam, usually split across one 2-mark short-answer question on order/degree or forming a DE, and one 5-mark long-answer question requiring a full solution method (variable separable, homogeneous, or linear). The chapter has featured in every CBSE Class 12 Maths paper in recent years.
What is the difference between the order and degree of a differential equation? +
The order of a differential equation is the order of the highest-order derivative present. The degree is the power of that highest-order derivative, provided the equation is a polynomial in its derivatives. If the DE involves terms like sin(dy/dx) or e^(d²y/dx²), the degree is not defined. CBSE board questions frequently ask students to state both the order and degree of a given DE.
When should students use the variable separable method versus the linear DE method? +
Use the variable separable method when the DE can be written as f(x)dx = g(y)dy — i.e., all x terms (with dx) can be separated to one side and all y terms (with dy) to the other. Use the linear DE method when the equation has the form dy/dx + P(x)y = Q(x) or dx/dy + P(y)x = Q(y), which requires finding the integrating factor e^(∫P dx). The homogeneous method applies when both sides of dy/dx = F(x,y) are homogeneous functions of the same degree.
What is an integrating factor and why is it important for CBSE board exams? +
An integrating factor (IF) is a function, typically e^(∫P dx), multiplied to both sides of a linear DE to make the left-hand side an exact derivative of the form d/dx[y · IF]. After multiplying, the equation integrates directly. CBSE board 5-mark questions on linear DEs almost always require students to correctly identify P(x), compute the integrating factor, and then integrate — errors in the IF are the most common reason for lost marks in this chapter.
How do I generate a custom question paper for Differential Equations using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 12 Mathematics, select Chapter 9 (Differential Equations), set your preferred question-type mix (MCQ, short answer, long answer) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.