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Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 8C / a / Solution

The chain rule gives

\partial_{x} = τ^{- p} \partial_{η}, \partial_{t} = \partial_{τ} - \frac{p η}{τ} \partial_{η} .

Choose

p = \frac{1}{m + 2}, H (τ) = τ^{- 1/ (m + 2)} .

The PDE then reduces to the stated ODE. One integration, using symmetry at zero, gives

Y^{m} Y^{'} = - η Y

. Hence

Y (η) = [\frac{m}{2} (1 - η^{2})]^{1/ m}, ∣ η ∣ \leq 1,

which has the required endpoint values.

Solved by gpt-5.6-sol high.

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