= Solution
The <Leibniz rule> rewrites the left side as $(xu)'$. By <a distribution with zero derivative is constant>,
$$
(xu)'=\delta'_0\quad\Longrightarrow\quad xu=\delta_0+c.
$$
The <Dirac delta multiplication identity> $x\delta'_0=-\delta_0$ supplies a particular solution, and the preceding division result supplies all solutions of $xu=c$. Hence
$$
\boxed{u=-\delta'_0+c\operatorname{pv}\frac1x+d\delta_0,\qquad c,d\in\mathbb C.}
$$
Multiplying by $x$ and then differentiating verifies the equation. The coordinate-multiplication kernel and the constant-derivative kernel show that no additional solutions are missing.
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