= Solution
Substitute the three nonzero initial-<velocity> coefficients into the general <Fourier sine series>, with $\omega_1=\pi c/L$:
$$
\boxed{y(x,t)=\frac{V_0}{\omega_1}\left[
\sin\frac{\pi x}{L}\sin\omega_1t+
\frac14\sin\frac{2\pi x}{L}\sin2\omega_1t+
\frac29\sin\frac{3\pi x}{L}\sin3\omega_1t\right].}
$$
Every term satisfies the <wave equation> and the endpoint conditions. Differentiating in time at zero recovers the three coefficients $V_0,V_0/2,2V_0/3$, providing a direct initial-data check.
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