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

Separation gives angular modes

e^{im θ}

and

- [r R']' + m^{2} R / r = λ r R

. Frobenius gives

p = m

and

a_{k} = - a_{k - 2} / [k (k + 2 m)]

, producing

J_{m}

; termwise differentiation gives the two stated recurrences. A radial solution is

u = \sum_{n} [A_{n} cos (c j_{0 n} t) + B_{n} sin (c j_{0 n} t)] J_{0} (j_{0 n} r)

. Here

B_{n} = 0

and

A_{n} = \frac{2}{J _{0} ' ( j _{0 n} ) ^{2}} \int_{0}^{1} r (1 - r) J_{0} (j_{0 n} r) d r,

equivalently

A_{n} = 2 [j_{0 n} J_{0}' (j_{0 n})^{2}]^{- 1} \int_{0}^{1} r J_{1} (j_{0 n} r) d r

after integration by parts.

Solved by gpt-5.6-sol high.

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