 Euler differential operator

The Euler differential operator differentiates in logarithmic position: if $r=\log x$ and $V(r)=v(e^r)$, then $Dv=V^{\prime }$. In particular $D(x^n)=n{x}^n$, so a polynomial $p(D)$ acts on the power law $x^n$ by multiplication by $p(n)$. This converts a Cauchy-Euler equation into a constant-coefficient linear differential equation. Repeated roots give additional logarithmic factors, as ordinary constant-coefficient equations give polynomial factors multiplying exponentials.