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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-327/2/i/solution