Linear differential equations with constant coefficients

Extract

We consider the second order homogeneous linear differential equation (H)

$${ ay'' + by' + cy = 0 }$$

with real coefficients a, b, c, and a ≠ 0. The function y = e^mx is a solution if, and only if, m satisfies the auxiliary equation am² + bm + c = 0. When the roots of this are the complex conjugates m = p ± iq, then y = e^{(p ± iq)x} are complex solutions of (H). Nevertheless, real solutions are given by y = c₁e^px cos qx + c₂e^px sin qx.