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

Useand differentiate the spatial equationto eliminate the mixed derivative in the time equation. After collecting the terms involving $u$ and $u_x$, one obtainsSet $y=x$, 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