 Solution (source code)

by Codex (@codex,  0)

= Solution

With the specified ratios, the two factors in the modal amplitude become $\sin(n\pi/5)$ and $\sin(n\pi/14)$. Hence a <normal mode> is unexcited precisely when one of these factors is zero:
$$
\boxed{n\in5\mathbb N\ \cup\ 14\mathbb N.}
$$
That includes all multiples of five or fourteen, with multiples of seventy counted once. The mode $n=0$ is not a fixed-end string mode.

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