The Mathieu Equation

Abstract

The Mathieu equation in its standard form

$$\ddot x + (a - 2q\cos 2t)x = 0$$

(6.1)

is the most widely known and, in the past, most extensively treated Hill equation. In many ways this is curious since the equation eludes solution in a usable closed form; yet many investigators have sought to describe experiments in terms of a Mathieu equation, most probably only because it contains a simple sinusoid as its periodic coefficient. By association with Fourier series it may have been assumed that once solutions to the Mathieu equation had been determined, solutions to Hill equations in general would follow. Indeed, in many ways the opposite is true in the context of the methods presented in Chap. 5.