= Solution
{c}
Use the same difference $w$ and <wave energy> as in part (i). The <damped wave equation> now gives the <energy dissipation identity for a linearly damped string>
$$
\boxed{E'(t)=\int_0^\pi w_t(w_{tt}-w_{xx})\,dx
=-2\alpha\int_0^\pi w_t^2\,dx\leq0.}
$$
Because $E(0)=0$ and $E(t)\geq0$, the <energy> remains zero. Hence $w_t=w_x=0$, and the homogeneous <Dirichlet boundary conditions> force $w=0$. Therefore the <solution of a differential equation> is unique also for $\boxed{\alpha>0}$; positive <damping> can only decrease this <energy>.
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