Distributional Solutions to $xu'=0$

Question

I want to solve the differential equation $xu' = 0$ for some distribution $u \in \mathcal{D}(\mathbb{R}^n)$. It seems apparent that the solution $u$ should be of the form $u = c_1 + c_2H$ where $c_1,c_2$ are constants, and $H$ is the Heaviside function. Is there any way I can prove that these are the only solutions rigorously?