Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 13A / i / Solution

LetTheir Bessel equations in self-adjoint form areMultiply the first by $v$, the second by $u$, subtract, and integrate. The left side becomes a boundary term:Regularity at zero makes the lower boundary term vanish, whileTherefore

If $\alpha ={\gamma }_k$ and $\beta ={\gamma }_{\ell }$ with $k\ne \ell$, both endpoint values of $J_0$ vanish, so the integral is zero. This is the weighted orthogonality of distinct eigenfunctions in a Sturm-Liouville problem. For the norm, hold $\alpha ={\gamma }_k$ and let $\beta \to {\gamma }_k$. Sincethe quotient tends to