The ODE $xy' + y = 0$ has no strong solution over $\mathbb R$ but has solution $y(x) = \begin{cases} c_1/x & x<0 \\ c_2/x & x>0 \end{cases}$ over $\mathbb R^*$, which may equivalently be written as $y(x) = \frac ax + \frac b{|x|}, x\neq 0$. In Friedlander's "Introduction to the theory of distributions" exercise 2.3 page 30, the corresponding distributional differential equation $$ x T'_x + T_x = 0 $$ where $T$ is a distribution (generalized function) and $x T'_x$ is the distribution $\langle xT'_x, \phi(x)\rangle = \langle T'_x, x \phi(x)\rangle $, is shown to have the solution $$ T_x = a\,\text{p.v.}\frac1x + b\, \delta(x) $$ where $\text{p.v.}\frac1x$ is the Cauchy principal value distribution, and $\delta$ is Dirac's delta distribution.
I am a little bothered that the strong and weak solutions are a little different. I understand why $\delta$ appears due to the singularity at $x = 0$, but I don't quite see why we lose the $\frac1{|x|}$ term, which might have been defined as a Hadamard finite part distribution?