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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 34D / d / Solution

Use

(\partial_{x} + \partial_{y})^{3} = \partial_{x}^{3} + \partial_{y}^{3} + 3 \partial_{x} \partial_{y} (\partial_{x} + \partial_{y})

and differentiate the spatial equation

(\partial_{x}^{2} - \partial_{y}^{2}) K = u K

to eliminate the mixed derivative in the time equation. After collecting the terms involving

u

and

u_{x}

, one obtains

K_{t} + (\partial_{x} + \partial_{y})^{3} K - 3 u (\partial_{x} + \partial_{y}) K = 0 .

Set

y = x

, use

u = - 2 \partial_{x} K (x, x, t)

with the derivative along the diagonal, and differentiate once more in

x

. The resulting compatibility condition is

u_{t} + u_{xxx} - 6 u u_{x} = 0 .

Solved by gpt-5.6-sol high.

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