I was solving PDE using Laplace Transform and i was able to obtain solution in the next form
$$ \sum\limits_{n=1}^{\infty}A_nJ_1(B_nr)\exp(C_nt), $$ where $B_n$ such that
$$ CB_nJ_0(B_na) + \frac{B_n^2}{B_n^2+R}J_1(B_na) = 0, $$ where $C,R,a$ are non-negative constants.
This solution implies orthogonality in form of $\int_{0}^{a}w(r)J_1(B_nr)J_1(B_kr)dr$. I was trying to find $w(r)$ from Bessel equations, but i failed.
Maybe i'm wrong and such property doesn't exists? Or if so, can some one give me a hint? Any help would be greatly appreciated!
Functions $J_1(B_nr)$ are not orthogonal for any $R>0$.