Repeated-root constant-coefficient differential equation
= Repeated-root constant-coefficient differential equation
{title2=$(D-r)^2y=f$}
The homogeneous equation has <linearly independent> solutions $e^{rx}$ and $xe^{rx}$, with nonzero <Wronskian> $e^{2rx}$. Setting $y=e^{rx}u$ converts the forced equation to $u''=e^{-rx}f$. For resonant forcing $f=e^{rx}$, the particular response is $x^2e^{rx}/2$. This substitution explains the extra powers of $x$ required by repeated-root resonance.