 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

When $\alpha=n$, the reduced <ordinary differential equation> is $F''=0$. The <initial conditions> $F(0)=0$, $F'(0)=1$ give
$$
\boxed{F(t)=t,\qquad y(x,t)=te^{-nt}\sin(nx).}
$$
This is the <critically damped> <normal mode>, also obtained by taking $\nu\to0$ in $\sin(\nu t)/\nu$.

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