Homogeneous equation and characteristic equation

General form

We consider the equations ay'' + by' + c*y = 0, where a, b and c are real numbers and a is not equal to 0. This is a homogeneous second-order linear equation with constant coefficients.

Characteristic equation

We are looking for solutions of the form y = exp(rx). Substituting this into the equation gives: ar^2 + b*r + c = 0 This is the characteristic equation. Its discriminant, δ = b² - 4ac, determines the nature of the solutions.

The three cases

Discriminant	Roots	General solution
delta > 0	distinct real roots r1, r2	y = C1exp(r1x) + C2exp(r2x)
delta = 0	double root r0	y = (C1 + C2x) * exp(r0x)
delta < 0	r = p ± i*q (complex)	y = exp(px) * (C1cos(qx) + C2sin(q*x))

Example

Solve y'' - 3y' + 2y = 0. Characteristic equation: r² - 3r + 2 = 0, delta = 1, roots r = 1 and r = 2. General solution: y(x) = C1exp(x) + C2exp(2x)

Another example: y'' + 4y = 0. r² + 4 = 0, purely imaginary roots r = ±2i (p = 0, q = 2). Solution: y(x) = C1cos(2x) + C2sin(2x)

Common pitfall

For delta = 0, omitting the factor x in front of C2 is a common mistake: the double root imposes a ‘resonant’ solution xexp(r0x); otherwise, one loses a dimension of the solution space, which must be of dimension 2 for a second-order equation.