Second-order linear homogeneous ODE

ay'' + by' + cy = 0. Solve auxiliary equation aλ² + bλ + c = 0.

Three cases by discriminant: distinct real roots λ₁, λ₂ → y = Ae^λ₁x + Be^λ₂x; repeated root λ → y = (A + Bx)e^λx; complex roots p ± qi → y = e^px(A cos qx + B sin qx).

Worked examples

y'' − 5y' + 6y = 0: roots 2, 3; y = Ae^2x + Be^3x.

y'' + 4y = 0: roots ±2i; y = A cos 2x + B sin 2x. Simple harmonic motion.

y'' − 4y' + 4y = 0: repeated root 2; y = (A + Bx)e^2x.

Frequently asked questions

SHM?

Second-order ODE with imaginary roots gives sinusoidal solutions. The classic spring/mass equation.

Non-homogeneous?

Add particular integral to general solution.