Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods

In the study of ordinary differential equations (ODEs) of the form (L) over cap [y(x)] = f(x), where L is a linear differential operator, two related phenomena can arise: resonance, where f (x) proportional to u(x) and (L) over cap [u(x)] = 0, and repeated roots, where f(x) = 0 and (L) over cap = D-n for n >= 2. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution u(x), introducing a parameter epsilon such that u(x) -> u(x; epsilon), and Taylor expanding u(x; epsilon) about epsilon = 0. The coefficients of this expansion partial derivative(k)u/partial derivative epsilon(k) vertical bar(epsilon=0) yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. We provide examples of many common ODEs, including constant coefficient, equidimensional, Airy, Bessel, Legendre, and Hermite equations. While the ideas can be introduced at the undergraduate level, we could not find any existing elementary or advanced text that illustrates these ideas with appropriate generality.