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

Remove the common nonzero factor $e^{-i\omega t}$ and expand in the small parameter $ϵ$. To first order, the modified Laplacian contributes only its $y$ derivatives: its $x$ derivative term has the extra factor ${ϵ}^2$. The regular perturbation expansion therefore givesThus the two residuals are $P+ϵQ+O({ϵ}^2)$ and $R+ϵS+O({ϵ}^2)$, respectively. Subscripts after a comma denote partial derivatives. We retain $\omega \ne 0$, as appropriate to the oscillatory wave and the elimination below.