= Solution
Free ends give $Y'(0)=Y'(L)=0$, so the oscillatory <normal modes> are
$$
\boxed{k_n=\frac{n\pi}L,\qquad
y_n=A_n\cos\frac{n\pi x}L\cos(ck_nt+\phi_n),\qquad n=1,2,\ldots.}
$$
There is also the <zero mode> $k_0=0$, with uniform displacement $y_0(x,t)=a+bt$: a free string can translate without a restoring force. It is not a positive-frequency oscillation. The first three oscillatory shapes are $\cos(\pi x/L)$, $\cos(2\pi x/L)$ and $\cos(3\pi x/L)$, with an <antinode> at each end. The figure includes the uniform mode too, so that the zero-frequency limit is explicit.
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