I'm trying to prove
$$\int_0^1xJ_n(ax)^2\,dx=\frac{1}{2}{J_n'(a)}^2$$
where $a$ is a zero of the nth Bessel function. The furthest I am getting is substituting $z=ax$ and then integrating by parts to find
$$I=-\frac{1}{a^{2}}\int_0^az^2J_n(z)J_n'(z)dz$$
A hint I've been given is to use the fact that $J_n$ satisfies the Bessel equation but I can't find a way to do it. I'd love to see how to do it this way if possible. I know there's already another solution here Bessel function of first kind orthogonality equation but I'd like to see if it is possible to do as the hint describes? Thank you in advance for your help!