Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 2 / 14D / Solution

Separation gives angular modes $e^{im\theta }$ and $-[rR\prime ]\prime +m^{2}R/r=\lambda rR$. Frobenius gives $p=m$ and $a_k=-a_{k-2}/[k(k+2m)]$, producing $J_m$; termwise differentiation gives the two stated recurrences. A radial solution is $u={\sum }_n[A_n\cos (c{j}_{0n}t)+B_n\sin (c{j}_{0n}t)]J_0(j_{0n}r)$. Here $B_n=0$ andequivalently $A_n=2[j_{0n}{J}_0\prime (j_{0n}{)}^2{]}^{-1}{\int }_0^{1}r{J}_1(j_{0n}r)dr$ after integration by parts.