Differential Equations

Why This Chapter Matters

Differential Equations give 5-7 marks in CBSE Class 12. Order, degree, variable separable method, and linear differential equations (integrating factor) are the standard board question types.

Core Concepts

1. Definitions

Differential equation: Equation involving derivatives of a function.

Order: Order of the highest derivative present.

Degree: Power of the highest order derivative (when DE is polynomial in derivatives).

Note: degree is NOT defined if sin(y'), eʸ', etc. appear.

Examples:

dy/dx + 2y = 0 → Order 1, Degree 1.

d²y/dx² + (dy/dx)³ = 0 → Order 2, Degree 1.

(d²y/dx²)³ + dy/dx = x → Order 2, Degree 3.

2. Variable Separable Method

If dy/dx = f(x)·g(y), separate variables:

dy/g(y) = f(x)dx, then integrate both sides.

Example: dy/dx = x²y

dy/y = x²dx → ln|y| = x³/3 + C → y = Ae^(x³/3).

3. Homogeneous Differential Equations

dy/dx = F(y/x) — degree of x and y same on both sides.

Substitution: y = vx → dy/dx = v + x(dv/dx).

Then separate variables in x and v.

Example: (x²+y²)dx - 2xydy = 0 → dy/dx = (x²+y²)/2xy.

Put v=y/x: v+x(dv/dx) = (1+v²)/2v.

x(dv/dx) = (1+v²)/2v - v = (1-v²)/2v.

2v dv/(1-v²) = dx/x → -ln|1-v²| = ln|x| + C → x(1-v²) = k → x-y²/x = k.

4. Linear Differential Equations (Most Important for Boards)

Form: dy/dx + P(x)y = Q(x)

Integrating Factor (IF): μ = e^(∫P dx)

Solution: y·μ = ∫Q·μ dx + C

Steps:

Example: dy/dx + y/x = x²

P=1/x, Q=x². IF = e^(∫1/x dx) = e^(ln x) = x.

Multiply: x(dy/dx) + y = x³ → d/dx(xy) = x³.

Integrate: xy = x⁴/4 + C → y = x³/4 + C/x.

5. Applications

Growth and Decay: dN/dt = kN → N = N₀eᵏᵗ.

Radioactive decay: k < 0.

Population growth: k > 0.

Newton's Law of Cooling: dT/dt = k(T-T₀) where T₀ = ambient temperature.

Solution: T-T₀ = (T₀₀-T₀)eᵏᵗ.

Board Examples

Q1: Solve: (x+1)dy/dx = 2xy.

Separate: dy/y = 2x/(x+1)dx = 2[1 - 1/(x+1)]dx.

Integrate: ln|y| = 2x - 2ln|x+1| + C.

y = A·e^(2x)/(x+1)².

Q2 (Linear): dy/dx - y/(x+1) = (x+1)².

P = -1/(x+1), Q = (x+1)². IF = e^(-∫dx/(x+1)) = e^(-ln(x+1)) = 1/(x+1).

d/dx[y/(x+1)] = (x+1). y/(x+1) = (x+1)²/2 + C. y = (x+1)³/2 + C(x+1).

PYQs (CBSE)

CBSE 2023: Find order and degree of: x²(d²y/dx²)³ + y(dy/dx)⁴ = 0.

Highest derivative: d²y/dx² (order 2). Its power = 3 (degree 3). Order=2, Degree=3.

CBSE 2022: Solve the DE: dy/dx = (y/x) + tan(y/x).

Homogeneous. y=vx → v+x(dv/dx) = v + tanv → x(dv/dx) = tanv.

dv/tanv = dx/x → cosv/sinv dv = dx/x → ln|sinv| = ln|x| + C.

sinv = kx → sin(y/x) = kx.

Revision Notes

ORDER: highest derivative's order
DEGREE: power of highest order derivative (if polynomial)
Degree undefined if: sin(y'), eʸ', log(y'') etc.

VARIABLE SEPARABLE: dy/dx=f(x)g(y) → dy/g(y)=f(x)dx → integrate

HOMOGENEOUS: degree same → put y=vx, dy/dx=v+x(dv/dx)

LINEAR: dy/dx + Py = Q
IF = e^(∫P dx)
Solution: y·IF = ∫Q·IF dx + C

APPLICATIONS:
Growth/Decay: N=N₀eᵏᵗ
Newton cooling: T-T₀=(Ti-T₀)eᵏᵗ