Last Updated: May 2026
Differential Equations (NCERT Class 12 Mathematics, Chapter 9) is a high-yield CBSE board chapter — typically 8–10 marks across short-answer and long-answer questions. The chapter rewards three skills: (1) identifying the type of differential equation, (2) selecting the correct solving method, (3) applying integrating factor for linear DEs.
Quick Reference Table — Methods of Solution
| Type | Identifier | Method |
|---|---|---|
| Variables Separable | Can be written as f(y)dy = g(x)dx | Integrate both sides |
| Homogeneous | dy/dx = f(y/x) | Substitute y = vx |
| Linear (in y) | dy/dx + Py = Q where P, Q are functions of x only | Multiply by IF = e^∫P dx |
| Linear (in x) | dx/dy + Px = Q where P, Q are functions of y only | Multiply by IF = e^∫P dy |
| Exact | M dx + N dy = 0 with ∂M/∂y = ∂N/∂x | (NCERT covers solution by inspection only) |
Order and Degree — The Definitions
- Order = order of the highest derivative occurring in the DE
- Degree = power of the highest-order derivative after the equation is freed from radicals and fractions of derivatives
Caution: Degree is undefined if the DE is not a polynomial in derivatives. NCERT explicitly excludes such cases from board questions.
NCERT Solutions — Variables Separable Examples
Example: Solve dy/dx = (1 + y²) / (1 + x²).
Solution: Separate: dy/(1+y²) = dx/(1+x²). Integrate: tan⁻¹y = tan⁻¹x + C.
Example: Solve sec²x · tan y dx + sec²y · tan x dy = 0.
Solution: Divide by tan x · tan y: (sec²x / tan x) dx + (sec²y / tan y) dy = 0. Integrate: log|tan x| + log|tan y| = log C → tan x · tan y = C.
Homogeneous Differential Equations
Definition: A function f(x, y) is homogeneous of degree n if f(λx, λy) = λⁿ · f(x, y). A DE dy/dx = f(x, y)/g(x, y) is homogeneous if f and g are both homogeneous of the same degree.
Standard substitution: y = vx, dy/dx = v + x dv/dx. This converts the DE into variables separable form in v and x.
Example: Solve (x² − y²) dx + 2xy dy = 0.
Solution: Divide: dy/dx = (y² − x²)/(2xy). Substitute y = vx: v + x dv/dx = (v²x² − x²)/(2x · vx) = (v² − 1)/(2v). Rearrange: x dv/dx = (v² − 1)/(2v) − v = −(v² + 1)/(2v). Separate and integrate.
Linear Differential Equations — The Integrating Factor
Standard form: dy/dx + Py = Q, where P and Q are functions of x.
Three steps:
- Compute integrating factor IF = e^∫P dx
- Multiply the entire DE by IF — the LHS becomes d/dx (y · IF)
- Integrate both sides: y · IF = ∫(Q · IF) dx + C
Example: Solve dy/dx + y = e⁻ˣ.
Solution: P = 1, IF = e^∫1 dx = eˣ. Multiply: eˣ dy/dx + eˣ y = e⁰ = 1. So d/dx (y eˣ) = 1. Integrate: y eˣ = x + C → y = (x + C) e⁻ˣ.
Important Questions — Past CBSE Boards Pattern
- Find order and degree of (d²y/dx²)³ + 2(dy/dx)² − y = 0 → Order 2, Degree 3
- Solve dy/dx = (x + y) /x → homogeneous; substitute y = vx
- Solve (1 + x²) dy/dx + 2xy = 1/(1+x²) → linear; IF = e^∫2x/(1+x²) dx = 1+x²
- General solution of dy = sin x dx → integrate, y = −cos x + C
- Particular solution given initial condition → substitute the initial values into the general solution to find C
Application Problem — Population Growth
Population P grows at rate proportional to P: dP/dt = kP. This is variables separable: dP/P = k dt → log P = kt + C → P = P₀ e^(kt). Same form models radioactive decay (k negative), Newton’s law of cooling, and bank-account compound interest.
Practice MCQs — CBSE Class 12 Differential Equations
Quiz data missing.
Frequently Asked Questions
When is the degree of a differential equation undefined?
When the equation is not a polynomial in its derivatives — for example when a derivative occurs inside a sin, log or exponential function, or as a fractional power that cannot be cleared. NCERT board questions exclude such DEs.
How do I identify a homogeneous DE?
Express dy/dx and check whether it can be written purely as a function of (y/x). If yes, the equation is homogeneous and the substitution y = vx solves it.
What is the integrating factor of dy/dx + (1/x)y = sin x?
P = 1/x. IF = e^∫(1/x) dx = e^(log x) = x. Multiply through: x dy/dx + y = x sin x → d/dx(xy) = x sin x. Integrate by parts: xy = sin x − x cos x + C.
What is the difference between general solution and particular solution?
A general solution contains an arbitrary constant C. A particular solution is obtained by substituting given initial or boundary conditions to determine the value of C.
Continue Your Class 12 Maths Prep
- CBSE Class 12 Maths — Continuity and Differentiability
- Ready For Boards Courses
- Board Exam FAQ
- CBSE Mock Test
Bottom line: Identify the type (separable, homogeneous, linear), apply the right method, and master the integrating-factor pattern. Differential equations contribute roughly 8–10 marks every CBSE board paper — guaranteed marks for the formula-fluent student.
