  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.
