Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2 / f / Solution

Take the Fourier transform in $x$, with convention $\tilde{v}(k_x)=\int v(x)e^{-i{k}_{x}x}dx$. The differentiation rules $\widehat{v^{\prime \prime }}=-k_x^2\tilde{v}$ and $\widehat{x^{2}v}=-{\tilde{v}}^{\prime \prime }$ turn the forced equation intoFor the unforced equation, $d{k}_x/dt=S{k}_y$ impliesIt is therefore exactly the shearing-wave oscillator found in part c, now parametrized by radial wavenumber rather than time.