Exponential shift rule for differential operators
= Exponential shift rule for differential operators
{title2=$P(D)[e^{ax}v]=e^{ax}P(D+a)v$}
For a constant $a$ and a <polynomial> $P$, the <product rule> gives $D(e^{ax}v)=e^{ax}(D+a)v$. Iteration and <linearity> prove the displayed identity. In particular $(D-a)^m(e^{ax}v)=e^{ax}v^{(m)}$, which reduces a repeated-root constant-coefficient <linear ordinary differential equation> to repeated <integration>.