Past exam of the mathematical methods II course of the University of Oxford / 2022 / 12 / a / Solution

Use separation of variables, $y=X(x)T(t)$. A nontrivial separated solution satisfies $X^{\prime \prime }/X=T^{\prime \prime }/(c^{2}T)=-k^2$. Fixed endpoints force $X(0)=X(L)=0$, so $X=\sin (n\pi x/L)$, $n=1,2,\dots$. ThereforeHere $c>0$ is the wave speed. Zero or negative $k^2$ gives no nonzero fixed-end spatial solution. Each time coefficient is a simple harmonic oscillator.