 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/2/b/solution

u = cos θ

. The boundary data simplify to

u + 2 u^{2} = \frac{2}{3} P_{0} (u) + P_{1} (u) + \frac{4}{3} P_{2} (u)

. The axisymmetric harmonic extension from a sphere keeps only decaying radial powers outside and regular powers inside. Hence

Ψ_{out} = \frac{2 a}{3 r} + \frac{a ^{2}}{r ^{2}} cos θ + \frac{4 a ^{3}}{3 r ^{3}} P_{2} (cos θ), r \geq a,

and

Ψ_{in} = \frac{2}{3} + \frac{r}{a} cos θ + \frac{4 r ^{2}}{3 a ^{2}} P_{2} (cos θ), r \leq a .

r = a

. The exterior solution tends to zero, and the interior solution is finite at

r = 0

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