$m$-order Partial Differential Equation General Solution

Question

Show that if $E$ is a solution of an $m$-order partial differential equation then $P(\partial)u=\sum\limits_{n=0}^ma_n\partial^nu=\delta$ where $\delta$ is the Dirac delta function, then $E\ast f$ is the solution of the partial differential equation $P(\partial)u=f$, where $f$ is the convolution.

I need some help getting started on this one, not really sure where to begin.

Answer 1

A possible solution is as follows:

By hypothesis

$P(\partial)E==\delta$

Then

$P(\partial)(E\ast f)=(P(\partial)E)\ast f = \delta \ast f = f $

Do you agree?