= Solution
Substituting $y(x,t)=f(x)g(t)$ into the <wave equation on a string> gives $f''g=fg''/c^2$. Wherever both factors are nonzero,
$$
\frac{f''(x)}{f(x)}=\frac{g''(t)}{c^2g(t)}.
$$
The left side depends only on $x$ and the right side only on $t$, so both equal a <separation constant> $\alpha$. Therefore
$$
\boxed{f''=\alpha f,\qquad g''=\alpha c^2g.}
$$
The resulting differential equations extend through zeros of $f$ or $g$ by continuity; division at a node is unnecessary.
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