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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 33D / c / Solution

The flow of

x \partial_{x} - α y \partial_{y}

is

G_{ε} : (x, y, u) ⟼ (e^{ε} x, e^{- α ε} y, u) .

Under the induced action on solutions,

u_{x y}

acquires the factor

e^{(α - 1) ε}

, while

e^{u} - e^{- 2 u}

is unchanged. The equation is invariant for every

ε

exactly when

α = 1.

Solved by gpt-5.6-sol high.

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