 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-2/32c/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Differentiating

L = - \frac{d}{d x} p^{2} \frac{d}{d x}

with respect to time gives

L_{t} = - \frac{d}{d x} (2 p p_{t} \frac{d}{d x}) .

p_{t} = p^{3} p_{xxx}

, then

L_{t} = - \frac{d}{d x} (2 p^{4} p_{xxx} \frac{d}{d x}) = L A - A L .

Thus $L_{t} = [L, A]$ is an Isospectral Lax equation, and $(L, A)$ is a Lax pair for the Harry Dym equation.

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