Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 13A (Methods)

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

By separation of variables, the axisymmetric normal modes satisfying regularity at $r=0$ and the fixed-edge condition at $r=1$ arewhere $J_0({\gamma }_k)=0$. The zero initial displacement removes all cosine terms, soDifferentiating at $t=0$ gives the weighted Fourier-Bessel expansionMultiply by $r{J}_0({\gamma }_{\ell }r)$ and integrate from zero to one. The orthogonality and norm from part (i) giveUsingthe remaining integral isHenceand therefore