Chapter 9 – Differential Equations

General Form of Differential Equations

F(x, y, dy/dx, d²y/dx², ..., dⁿy/dxⁿ) = 0

Order & Degree of Differential Equations

Order : Highest order of derivative
Degree : Power of highest order derivative
Example:
(d²y/dx²)³ + (dy/dx) + 2 = 0
Order: 2 , Degree: 3

Variable Separable Differential Equations

General Form :
dy/dx = f(x)g(y)
Separate Variables:
dy/g(y) = f(x)dx
Integrate Both Sides :
∫(1/g(y))dy = ∫f(x)dx + C

Homogeneous Differential Equations

General Form :
dy/dx = f(y/x)
Put :
y = vx, where v is a function of x
Then:
dy/dx = v + x(dv/dx)

Linear Differential Equations

General Form:
dy/dx + P(x)y = Q(x)
Integrating Factor (I.F.):
I.F. = e ^∫P(x)dx
Solution Formula:
y(I.F.) = ∫(Q(x) * I.F.)dx + C

Differential Equations