Solution
= Solution
For <separation of variables>, take $u(x,t)=X(x)T(t)$ in the <damped wave equation>. Dividing by $c^2XT$, with $c>0$, gives $X''/X=(T''+\alpha T')/(c^2T)=-k^2$. The separated <ordinary differential equations> are therefore
$$
\boxed{X''+k^2X=0,\qquad T''+\alpha T'+c^2k^2T=0.}
$$
The negative separation constant anticipates the positive spatial <eigenvalues> selected by the fixed endpoints.