Exponential shift rule for differential operators ($P(D)[e^{ax}v]=e^{ax}P(D+a)v$)

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.