Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 2 / c / Solution

The first-order condition $Q=0$, combined with $b_0=-i{\psi }_{0,y}$, givesDifferentiate this expression with respect to $y$ and insert it into $S=0$. The ${\psi }_1$ terms cancel, leavingThus the compatibility partial differential equation isIt is exactly ${\partial }_y(2{\psi }_{0,x}+i{\psi }_{0,yyy})=0$. Integrating with respect to $y$ therefore giveswhere the integration constant may be any function of $x$. Conversely, differentiating this last equation with respect to $y$ recovers the compatibility equation.