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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 13B / b / Solution

The separated axisymmetric solution is

Φ (r, x) = \sum_{ℓ = 0}^{\infty} (A_{ℓ} r^{ℓ} + B_{ℓ} r^{- ℓ - 1}) P_{ℓ} (x) .

Since

x (1 - x^{2}) = \frac{2}{5} (P_{1} - P_{3})

, regularity at the origin and the boundary condition give

Φ (r, x) = \frac{2}{5} [\frac{r}{R} P_{1} (x) - (\frac{r}{R})^{3} P_{3} (x)] .

Solved by gpt-5.6-sol high.

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