Show that $J_0(x) H(x)$ is also a Bessel Equation solution

Question

Recall that $J_0(x)$ is a solution of the Bessel equation. d^2y/dx^2 + (1/x)dy/dx + y(x) = 0 . Show that $J_0(x) H(x)$ is also a solution, where $H(x)$ is the Heaviside step function.

Answer 1

With $$y(x)=J_0(x) H(x)$$ (where $J_0$ is the smooth solution of $u''+\frac1x u'+u=0,u(0)=1$)

then $$y' = J_0' H+J_0H'= J_0' H + \delta$$ so that $\frac1x y' $ and hence $y''+\frac1xy'+y$ is not a distribution, you meant $$x^2 y'' + x y'+x^2 y=0$$ which is true in the sense of distributions