Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 2 / 16A / b / Solution

The homogeneous solutions $(C_1+C_{2}x)e^{-x}$ are not $2\pi$-periodic unless $C_1=C_2=0$, so there is a unique periodic solution. For a forcing term $a_n\cos (nx)$, a trial solution $A_n\cos (nx)+B_n\sin (nx)$ givesUsing the Fourier series from part (a), for which only odd $n$ occur and $a_n=-4/(\pi n^2)$, yieldsThe coefficients decay fast enough to differentiate the series twice term by term, verifying the equation and periodicity.