Chapter 9 Differential Equations

Notes

Differential Equation ka meaning:

Jis equation me derivative jaise dy/dx, d2y/dx2 present ho, usse differential equation kehte hain.

Example:

dy/dx = x

d2y/dx2 + y = 0

Order:

Differential equation me highest order derivative ka order hi equation ka order hota hai.

Example:

d2y/dx2 + dy/dx + y = 0

Order = 2

Degree:

Jab differential equation derivatives ke polynomial form me ho, tab highest order derivative ki power degree hoti hai.

Example:

(d2y/dx2)3 + dy/dx = 0

Degree = 3

Degree tab define nahi hoti jab derivative root, fraction, sin, cos, exponential ke andar ho.

General solution:

Jisme arbitrary constant C hota hai, use general solution kehte hain.

Example:

dy/dx = x

Solution:

y = x2/2 + C

Particular solution:

Jab value of C initial condition se mil jaye, to particular solution milta hai.

Example:

y = x2/2 + C

If y = 1 at x = 0

Then C = 1

Particular solution:

y = x2/2 + 1

Variable separable method:

Agar equation ko aise likh sake:

f(y) dy = g(x) dx

Then dono side integrate kar do.

Example:

dy/dx = x/y

Step:

y dy = x dx

Integrate:

y2/2 = x2/2 + C

Linear differential equation:

Standard form:

dy/dx + Py = Q

Yahan P and Q x ke functions hote hain.

Solution method:

Step 1:

Equation ko dy/dx + Py = Q form me lao.

Step 2:

I.F. find karo.

I.F. = e^∫P dx

Step 3:

Solution formula:

y × I.F. = ∫Q × I.F. dx + C

Important:

Agar equation dx/dy + Px = Q form me ho, to variable y ke respect me solve karte hain.

Homogeneous differential equation:

Agar dy/dx = F(y/x) form me ho, to homogeneous equation hoti hai.

Method:

Put y = vx

Then:

dy/dx = v + x dv/dx

Then variables separate karke solve karo.

Common formulas:

∫x^n dx = x^(n+1)/(n+1), n not equal to -1

∫1/x dx =…