Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 20

The chain rule for the new coordinates gives ${\partial }_x={\partial }_{\xi }+{\partial }_{\eta }$ and ${\partial }_y={\partial }_{\xi }-{\partial }_{\eta }$. Since the coordinates depend linearly on $x,y$, their coefficients are constant. Expanding the squares of these differential operators givesHence the partial differential equation becomes $u_{\xi \eta }=1/4$. Integrating once in each variable introduces independent one-variable functions, giving the general solutionwhere $F,G$ are arbitrary twice-differentiable functions on the relevant characteristic intervals. Their sum includes an arbitrary additive constant.

The first boundary condition requires $u(x,0)=a{x}^2+d=0$ for all $x$, so $a=d=0$. Since $u_y=bx+2cy$, the second boundary condition gives $b=0$. The partial differential equation then reads $2a-2c=1$, fixing $c=-1/2$. Thus

On a smooth implicit branch, put $Q=x+z^2$ and require $Q\ne 0$. This is exactly the implicit function theorem condition, since the defining equation has vertical partial derivative $3Q$. Differentiating the cubic relation in $x$ and $y$ givesFor a second derivative, the dependence of $Q$ on $z$ must also be included: $Q_x=1+2z{z}_x$ and $Q_y=2z{z}_y$. ThusThe required cancellation follows immediately:The verification is local to smooth branches with $x+z^2\ne 0$; the implicit relation does not guarantee a differentiable function at its singular points.