= Resonant exponential particular solution
{title2=$y_p=xe^{rx}/P'(r)$}
For a constant-coefficient <linear differential equation> $P(D)y=e^{rx}$, where $D$ denotes differentiation, a simple root $P(r)=0$ forces a resonant trial. Factor $P(z)=(z-r)Q(z)$; since $(D-r)(xe^{rx})=e^{rx}$, commuting the differential operators gives $P(D)(xe^{rx})=Q(r)e^{rx}=P'(r)e^{rx}$. Thus the displayed expression is a <particular solution>. More generally, if the root has multiplicity $m$, the same factorization gives $P(D)(x^me^{rx})=P^{(m)}(r)e^{rx}$, so $y_p=x^me^{rx}/P^{(m)}(r)$. This is the extra polynomial factor required by <resonance in a differential equation>.
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