Linear differential operator ($L={\sum }_{j=0}^m{a}_j(x)D^j$)

A linear differential operator is a finite linear combination of derivatives with specified coefficient functions. It obeys $L(\alpha f+\beta g)=\alpha Lf+\beta Lg$ for scalar constants $\alpha ,\beta$. Variable coefficients need not commute with differentiation: $D(af)=a^{\prime }f+a{f}^{\prime }$.

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.