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

Insert the proposed leading relation into the two zeroth-order residuals. It givesand, since $b_{0,y}=-i{\psi }_{0,yy}$,Hence $\boxed{b_0=-i{\psi }_{0,y}⟹P=R=0}$. The same relation is forced by $P=0$ when $\omega \ne 0$; its $y$ derivative then automatically gives $R=0$.