 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/33d/d/solution

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

For

α = 1

, the invariant of the characteristic equations is

z = x y

, so an invariant solution has

u (x, y) = f (z)

. Then

u_{x y} = f^{'} (z) + z f^{''} (z) .

The invariant solutions are therefore characterized by

z f^{''} + f^{'} = e^{f} - e^{- 2 f} .

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