= Solution
Use <separation of variables> with a nontrivial product $\eta=XT$. Substitution into the <wave equation on a string> gives $\tau_0X''T=\lambda_0XT''$. Wherever the factors are nonzero,
$$
\frac{X''(x)}{X(x)}=\frac{\lambda_0}{\tau_0}\frac{T''(t)}{T(t)}.
$$
The left side depends only on $x$ and the right only on $t$, so both are a constant. For nontrivial fixed-end solutions this constant is negative: multiplying $X''=\sigma X$ by $X$ and integrating, using the homogeneous <Dirichlet boundary conditions>, gives
$$
\sigma\int_0^LX^2\,dx=-\int_0^L(X')^2\,dx<0.
$$
Write it as $-k^2$. The separated equations are therefore
$$
\boxed{X''+k^2X=0,\qquad T''+c^2k^2T=0.}
$$
These equations extend across the zeros of $X$ and $T$; the division is a derivation device, not a restriction excluding their nodes.
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