Consider the differential operator:
$D = \sum_{j = 0}^{r}\frac{d^j}{dx^j}$
And let $G_0$ be the solution for the homogeneous equation $DG_0 = 0$, satisfying the initial conditions:
$G_0(0)= 0, G_0^{(1)}(0) = 0,..., G_0^{(r-1)}(0) = \frac{1}{a_r}$
Then show that the distribution $G(x) = \theta(x)G_0(x)$ is a Green's function for D.
My attempt:
I used the product rule for distributions and expressed each derivative in terms of a finite sum, then expanded the product $\partial_{i}\theta \cdot \partial_{j-i}G_0$ into another sum, but this seems to be a really complicated and convoluted way to approach the problem. Is there maybe any other way someone could point me to?