Differential Equations — Mathematics Class 12 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Differential Equations (Class 12) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Differential Equations (CBSE & HBSE)
Differential Equations carries 8-10 marks in CBSE. Questions cover order/degree, formation by eliminating constants, variable separable method, homogeneous equations, and linear first-order DEs with integrating factor.
Basic Concepts — Order, Degree and Formation
Differential Equations — Key Terminology
Differential Equation (DE): An equation involving derivatives of a function.
Order: The order of the highest derivative in the equation. Degree: The power of the highest-order derivative (after clearing fractions/radicals).
| DE | Order | Degree |
|---|---|---|
| dy/dx + y = sin x | 1 | 1 |
| d²y/dx² + (dy/dx)² = 0 | 2 | 1 |
| (d²y/dx²)³ + dy/dx = x | 2 | 3 |
| √(dy/dx) = x+1 ⟹ dy/dx = (x+1)² | 1 | 2 (after squaring) |
Note: Degree is defined only when the DE is a polynomial in derivatives.
General and Particular Solutions
- General Solution: Contains arbitrary constants equal to the order of the DE.
- Particular Solution: Obtained by assigning specific values to arbitrary constants using initial/boundary conditions.
Formation of DE by Eliminating Arbitrary Constants
Method: Differentiate the function as many times as there are arbitrary constants, then eliminate the constants.
Example: Form DE for y = Ae^(2x) + Be^(3x)
- y' = 2Ae^(2x) + 3Be^(3x)
- y'' = 4Ae^(2x) + 9Be^(3x)
- Eliminate A and B: y'' − 5y' + 6y = 0
CBSE Tip: Number of arbitrary constants = order of the DE formed. Always eliminate ALL constants to get the final differential equation.
Variable Separable and Homogeneous Differential Equations
Variable Separable Method
Form: dy/dx = f(x)·g(y) Method: Separate variables — put all y terms with dy, all x terms with dx: dy/g(y) = f(x)dx Integrate both sides.
Example: Solve dy/dx = y sin x
- dy/y = sin x dx
- ∫dy/y = ∫sin x dx
- ln|y| = −cos x + C
- y = Ae^(−cos x) (A = e^C)
Homogeneous Differential Equations
Form: dy/dx = F(y/x) or dy/dx = f(x,y)/g(x,y) where f,g are homogeneous of same degree.
Method: Substitute y = vx (so dy/dx = v + x·dv/dx)
- Transform DE to separable form in v and x
- Solve the separable DE
- Substitute back v = y/x
Checking homogeneity: Replace x → λx, y → λy. If f(λx,λy) = λⁿf(x,y), it is homogeneous of degree n.
Example: Solve dy/dx = (x+y)/x = 1 + y/x
- Let y = vx: v + x·dv/dx = 1+v
- x·dv/dx = 1 ⟹ dv = dx/x
- v = ln|x| + C ⟹ y/x = ln|x| + C ⟹ y = x ln|x| + Cx
CBSE Tip: To identify a homogeneous equation, put x=1, y=t (or check if f(kx,ky)/f(x,y) = kⁿ). After substituting y=vx, the x should cancel out from the right side, leaving a purely separable equation in v.
Linear First-Order Differential Equations
Linear First-Order DE
Standard form: dy/dx + P(x)·y = Q(x)
Integrating Factor (IF): μ = e^(∫P(x)dx)
General Solution: y·μ = ∫Q(x)·μ dx + C
or equivalently: y = (1/μ)[∫Q(x)·μ dx + C]
Step-by-Step Method
- Identify P(x) and Q(x)
- Calculate IF = e^(∫P(x)dx) — do NOT include +C here
- Multiply both sides by IF: d/dx(y·IF) = Q(x)·IF
- Integrate both sides: y·IF = ∫Q(x)·IF dx + C
- Solve for y (if required)
Linear in x: dx/dy + P(y)·x = Q(y) IF = e^(∫P(y)dy), Solution: x·IF = ∫Q(y)·IF dy + C
Key Results and Examples
Example 1: Solve dy/dx + y = eˣ
- P=1, IF = e^(∫1·dx) = eˣ
- y·eˣ = ∫eˣ·eˣ dx = ∫e^(2x) dx = e^(2x)/2 + C
- y = eˣ/2 + Ce^(−x)
Example 2: Solve dy/dx − y tan x = 2 sin x
- P = −tan x, IF = e^(∫−tan x dx) = e^(ln|cos x|) = cos x
- y·cos x = ∫2 sin x·cos x dx = ∫sin 2x dx = −cos 2x/2 + C
- y·cos x = −cos 2x/2 + C
Applications of Linear DEs
- Growth/decay problems: dN/dt = kN
- Newton's Law of Cooling: dT/dt = k(T−T₀)
- RL circuits, mixing problems
CBSE Tip: The most common error in linear DEs is computing the wrong integrating factor. Remember: IF = e^(∫P dx) and P is the coefficient of y (not −y). If the equation is −y·P(x), identify P correctly after bringing to standard form.
Frequently asked questions
Are these Differential Equations notes free?
Yes — the Differential Equations notes for Mathematics (Class 12) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Differential Equations notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Differential Equations chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Differential Equations.