 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-3/16c/e/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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 .

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