Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 10 / b / i

Solution

\nabla^{2} ψ = \frac{1}{r ^{2}} \frac{d}{d r} (r^{2} \frac{d ψ}{d r}) .

Multiplying the Helmholtz equation by

r^{2}

gives

- \frac{d}{d r} (r^{2} ψ^{'}) = k^{2} r^{2} ψ, p (r) = r^{2}, q (r) = 0, w (r) = r^{2} .

The endpoint at

r = 0

is singular in Sturm-Liouville terminology; bounded regular radial fields select the physical solution and eliminate the

1/ r

branch. At

r = a

the field has the imposed zero Dirichlet value. For regular solutions and admissible trial functions, the endpoint term

r^{2} ψ ψ^{'}

vanishes at the origin and at the wall.

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