I would like to solve the distributional equation $$xu = f, $$ where $f \in C_c^\infty(\mathbb{R})$ and $u$ is a compactly supported tempered distribution on $\mathbb{R}$.
I have been told that the general solution to this equation is $$u = a \delta_0 + b \cdot \mathrm{p.v.}\frac{1}{x} + c, $$ where $a, b$ are constants and $c$ is some smooth function. However, I do not know how to prove this.
From the answer of this question How to solve distributional equations?, we have that the solution to my equation would be the sum of the general solution to the homogenous equation and the inhomogenous equation. The homogenous equation $$ xu = 0 $$ is solved by $a \delta_0$ where $a$ is a constant. However, I do not know how to continue from here.
Alternatively, by the Paley-Wiener theorem, we know that the Fourier transform of $u$ is a smooth function with some particular decay. Hence, if we take the Fourier transform of the equation, we get that $$i(\mathcal{F}u)' = \mathcal{F}f,$$ but I do not know how to solve this equation dsitributionally and conclude from here.