Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 9

For real coordinates, work on $y>0$. The chain rule gives ${\xi }_y=y^{-1/2}$ and ${\xi }_{yy}=-\frac{1}{2}{y}^{-3/2}$. ThereforeThe derivatives on the right are with respect to $\xi$. Behaviour at $y=0$ requires separate regularity of the composed function.

The same chain rule gives $u_{xx}=u^{\prime \prime }(\xi )$ and $u_y=u^{\prime }(\xi )/\sqrt{y}$. Substituting these and part (a) into the partial differential equation givesThus every twice-differentiable profile $u(\xi )$ gives a solution on $y>0$.