Differential Equations 🌊

Functions aur unki derivatives ke beech relationships — nature ki language!

🦠 Bacteria Growth

Ek petri dish mein bacteria hain. Har minute bacteria ki count double ho jaati hai — yaani growth rate population ke proportional hai. Math mein: dP/dt = kP (jahan k ek constant hai). Yeh ek differential equation hai! Iska solution P = P₀·eᵏᵗ hai — exponential growth. Yahi reason hai ki COVID ya koi bhi epidemic agar unchecked rahe toh exponentially spread hota hai. Differential equations nature ke sabse powerful tools hain.

1. Basic Definitions

Order

Highest derivative ka order. dy/dx → order 1. d²y/dx² → order 2.

Degree

Highest derivative ki power (jab DE polynomial form mein ho). (dy/dx)³ → degree 3.

General Solution

Arbitrary constants ke saath solution. n-th order DE → n arbitrary constants.

Particular Solution

Specific initial condition se constants determine ho jaate hain.

2. Variable Separable Method

Agar DE ko f(y)dy = g(x)dx form mein likh sako — toh dono sides integrate karo!

✅ Solve: dy/dx = y/x

Separate: dy/y = dx/x
Integrate: ln|y| = ln|x| + ln|C|
y = Cx (General Solution)

✅ Solve: dy/dx = e^(x+y), y(0)=0

e^(-y)dy = eˣ dx
-e^(-y) = eˣ + C → -e^(-y) = eˣ + C
y(0)=0: -1 = 1+C → C=-2. So eˣ + e^(-y) = 2

3. Homogeneous Differential Equations

DE ka form: dy/dx = F(y/x) (ratio y/x par depend karta hai)

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

Phir yeh variable separable ho jaata hai!

Put y = vx dy/dx = v + x(dv/dx) Substitute karo → separate variables in v and x Integrate → back substitute y = vx

4. Linear Differential Equations

Standard form: dy/dx + P(x)·y = Q(x) Integrating Factor (IF): μ = e^(∫P dx) Solution: y × IF = ∫(Q × IF)dx + C /* Also: dx/dy + P(y)·x = Q(y) — same method */

✅ Solve: dy/dx + y/x = x²

P = 1/x, Q = x². IF = e^(∫1/x dx) = e^(ln x) = x
y·x = ∫x²·x dx = ∫x³dx = x⁴/4 + C
y = x³/4 + C/x (General Solution)

5. Practice Questions

Q1. Solve: (1+x²)dy + 2xy dx = 0

Solution

dy/y = -2x dx/(1+x²). Integrate: ln y = -ln(1+x²) + ln C. y(1+x²) = C.

Q2. Solve: dy/dx = (x+y)/x where y(1)=1.

Solution

Homogeneous: v+x dv/dx = 1+v → x dv/dx=1 → dv=dx/x → v=ln x+C → y/x=ln x+C. y(1)=1: 1=0+C → C=1. y=x(1+ln x).

Q3. Solve: dy/dx - 3y = sin 2x

Solution

Linear DE. P=-3, IF=e^(-3x). y·e^(-3x) = ∫sin(2x)e^(-3x)dx. Use IBP twice. = e^(-3x)(-3sin2x-2cos2x)/13 + C. y = (-3sin2x-2cos2x)/13 + Ce^(3x).