Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 2 / c / Solution

Here $\lambda =7/2$ and ${\alpha }_n=n$. LetTwo applications of integration by parts giveSolving these equations,Consequently the required amplitudes areThe stated initial velocity is nonzero at the fixed endpoints, whereas a classical solution differentiable there would require zero endpoint velocity. Thus it is understood on the open interval, or as an L2 space initial datum in the usual finite-energy solution. Its Fourier sine series takes value zero at the endpoints; this changes no initial energy or interior initial data.