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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 16C / e / Solution

Set the similarity variable

ξ = \frac{r}{ν t}, ω (r, t) = W (ξ) .

Then

ω_{t} = - \frac{ξ}{2 t} W^{'}, ν (ω_{rr} + \frac{1}{r} ω_{r}) = \frac{1}{t} (W^{''} + \frac{1}{ξ} W^{'}) .

The factors of

t

cancel, leaving the self-similar ordinary differential equation

W^{''} + (\frac{1}{ξ} + \frac{ξ}{2}) W^{'} = 0 .

Solved by gpt-5.6-sol high.

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