# Find all distributions solving a differential equation

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.

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 $$x_2u=0$$.