 Solution

Use separation of variables with a nontrivial product $\eta =XT$. Substitution into the wave equation on a string gives ${\tau }_0{X}^{\prime \prime }T={\lambda }_{0}X{T}^{\prime \prime }$. Wherever the factors are nonzero,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 $X^{\prime \prime }=\sigma X$ by $X$ and integrating, using the homogeneous Dirichlet boundary conditions, givesWrite it as $-k^2$. The separated equations are thereforeThese equations extend across the zeros of $X$ and $T$; the division is a derivation device, not a restriction excluding their nodes.