 Solution

Substituting $y(x,t)=f(x)g(t)$ into the wave equation on a string gives $f^{\prime \prime }g=f{g}^{\prime \prime }/c^2$. Wherever both factors are nonzero,The left side depends only on $x$ and the right side only on $t$, so both equal a separation constant $\alpha$. ThereforeThe resulting differential equations extend through zeros of $f$ or $g$ by continuity; division at a node is unnecessary.