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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 327 / 2 / i / Solution

{c}

For smooth $a$ and a point $r$,
$$
a\delta_r=a(r)\delta_r,
\qquad a\delta'_r=a(r)\delta'_r-a'(r)\delta_r.
$$
These follow by applying both sides to a <test function> and using the <Leibniz rule>. In particular $(x-r)\delta'_r=-\delta_r$.

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 Incoming links: Dirac delta multiplication identity