 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-1/13b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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)] .

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