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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 9B / c / ii / Solution

The radial equation is

(r^{2} ϕ^{'})^{'} = - \frac{1}{r} .

Its solutions satisfying decay at infinity have the form

ϕ (r) = \frac{l o g r + 1 + C}{r} .

The boundary value

ϕ (1) = 1

gives

C = 0

, so

ϕ (r) = \frac{1 + lo g r}{r} .

It tends to zero as

r \to \infty

and directly satisfies

\nabla^{2} ϕ = - r^{- 3}

.

Solved by gpt-5.6-sol high.

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