I was studying some properties of regular Sturm-Liouville problems. I have seen that (given proper boundary conditions) given the problem
$$L(f)+ \lambda \omega f=0$$
With $L(f)=(rf')'+pf$ and $\omega, r,p $ regular real valued functions (not being too formal, sorry), then there are some properties which are satisfied. For instance, $\lambda$ is real and two eigenfunctions are orthogonal with respect to the scalar product $\int f \bar{g} \omega dx$.
Now, I have seen how to write a Bessel equation $f'' + \frac{f'}{x} + \frac{x^2 - n^2}{x^2} f=0$ as a Sturm-Liouville problem, which is
$$(xf')' - \frac{n^2}{x} f + xf$$
Where $p=-\frac{n^2}{x}$ and $\omega=x$. Now, this would mean that $\int J_n (ax)J_n(bx)xdx=0$ if $a \neq b$, and I have found this result on the internet as well. But I have been told (but maybe I just got it wrong and the result above was what he was trying to explain) that also $\int J_p(x) J_q(x)xdx=0$ if $p \neq q$, where the two functions are solutions of the Bessel equation with $n$ changed by $p$ and $q$. Now, I wasn't able to put this in a Sturm-Liouville form, because changing $n$ would change $L$ and I couldn't say anything. Also, I haven't found anything on the internet. So is there a way to prove it using Sturm-Liouville (and how can I do that) or I simply got it wrong and it isn't true?