Perturbing ordinary differential equations to generate resonant and repeated root solutions

TL;DR

The paper presents a novel perturbation method to generate exact solutions for resonant and repeated root cases in linear ordinary differential equations by expanding a known homogeneous solution with respect to a parameter.

Contribution

It introduces a perturbation approach using Taylor expansion of solutions to systematically derive resonant and repeated root solutions, offering an insightful alternative to traditional methods.

Findings

Provides a systematic perturbation method for resonance and repeated roots

Offers a less tedious alternative to reduction of order or variation of parameters

Applicable at undergraduate level with broad generality

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.