A second order homogeneous equation with constant coefficients

A second order homogeneous equation with constant coefficients is written as

where a, b and c are constant. This type of equation is very useful in many applied problems (physics, electrical engineering, etc..). Let us summarize the steps to follow in order to find the general solution:

Write down the characteristic equation

This is a quadratic equation. Let r₁ and r₂ be its roots (we have ). If r₁ and r₂ are distinct real numbers (this happens if b²-4ac > 0 ), then the general solution is

If r₁ = r₂ (which happens if b²-4ac = 0), then the general solution is

If r₁ and r₂ are complex numbers (which happens if b²-4ac < 0), then the general solution is

where

,

that is,

Example 1

Consider the differential equation y''(x) + y'(x) - 2y(x) = 0.

The characteristic equation is

r² + r - 2 = 0

so the roots are 1 and -2. That is, the roots are real and distinct. Thus the general solution of the differential equation is

y(x) = Ae^x + Be^-2x.

Example 2

Consider the differential equation y''(x) + 6y'(x) + 9y(x) = 0.

The characteristic equation has a repeated real root, equal to -3. Thus the general solution of the differential equation is

y(x) = (A + Bx)e^-3x.

Example 3

Consider the equation y''(x) + 2y'(x) + 17y(x) = 0.

The characteristic roots are complex. We have a = 2 and b = 17, so α = -1 and β = 4, so the general solution of the differential equation is

[A cos(4x) + B sin(4x)]e^-x.