Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 20 / a

Seek a separated solution $u=X(x)Y(y)$. On a region where the factors do not vanish, the partial differential equation separates asThe resulting ordinary differential equations integrate to $X=A{e}^{\lambda x}$ and $Y=B|3-y{|}^{\lambda /2}$ on either side of $y=3$. Choose $\lambda =2$ and the signed linear branch $Y=3-y$, which extends smoothly across that line. This yields the nonconstant solutionDirect partial differentiation gives $u_x=2e^{2x}(3-y)$ and $u_y=-e^{2x}$, whose prescribed combination vanishes everywhere. This direct check also covers the line on which the separated factor is zero.