Perturbing ordinary differential equations to generate resonant and repeated root solutions

Abstract

In the study of ordinary differential equations (ODEs) of the form $\hat{L}[y(x)]=f(x)$, where $\hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)\propto u(x)$ and $\hat{L}[u(x)]=0$, and repeated roots, where $f(x)=0$ and $\hat{L}=\hat{D}^n$ for $n\geq 2$. 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 $u(x)\rightarrow u(x;\epsilon)$, and Taylor expanding $u(x;\epsilon)$ about $\epsilon = 0$. The coefficients of this expansion $\frac{\partial^k u}{\partial\epsilon^k}\big{|}_{\epsilon=0}$ 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. While the ideas can be introduced at the undergraduate level, we could not find any elementary or advanced text that illustrates these ideas with appropriate generality.