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

For the linearly damped string, substitute $u(x,t)=X(x)T(t)$ and divide by $XT$:Thus separation of variables givesThe fixed-end Dirichlet boundary conditions require $X(0)=X(L)=0$. Nonzero solutions are $X_n(x)=\sin (n\pi x/L)$, with ${\alpha }_n=n\pi /L$, $n\ge 1$. Neither $\alpha =0$ nor a negative spatial eigenvalue produces a nonzero function obeying both endpoint conditions.