How to solve distributional equation?

Question

What are the only solutions of a distributional equation:

$$xT'=0$$

Thanks. Any hint?

I know that $T'(\phi)=-T(\Phi')$.

Answer 1

Hint. By definition of the product and derivative of distributions $$ 0=(xT')(\varphi)=T'(x\varphi)=-T\big((x\varphi)'\big)=-T(\varphi+x\varphi'), $$ for all $\varphi\in\mathscr D(\mathbb R)$.