Find all distributions solving a differential equation

Question

My problem is :

Find all distributions $u \in D'(\mathbb{R}^{2})$ such that $(x_1+ix_2)u=0$.

I know $c\delta$ is solution of $(x_1+ix_2)u=0$, but I am not sure if $c\delta$ is the general solution of it. If $c\delta$ is the general solution of $(x_1+ix_2)u=0$, I can't justify it.

I would like an anwer.

Answer 1

Hint Show that $x_1 u =0$ and $x_2u=0$. Deduce from here that $\sup(u) \subset \{ x_1=0\} \cap \{ x_2=0\}= \{ (0,0) \}$.

What can you say then about $u$?

Note If you learned about the Fourier transform, after you get that $u$ has compact support, take the FT of $x_1u=0$ and $x_2u=0$.