= Solution
The zero initial <displacement> selects sine time factors. Put $\omega_1=\pi c/L$. The <Fourier sine series> of the initial <velocity> has coefficients $V_1=V_0/2$, $V_2=-3V_0/4$, with all other coefficients zero. Since differentiation in time multiplies a sine by its <angular frequency>,
$$
\boxed{y(x,t)=\frac{V_0}{2\omega_1}\sin\left(\frac{\pi x}{L}\right)\sin(\omega_1t)-\frac{3V_0}{8\omega_1}\sin\left(\frac{2\pi x}{L}\right)\sin(2\omega_1t).}
$$
The <normal mode coordinates> are $q_1(t)=V_0\sin(\omega_1t)/(2\omega_1)$ and $q_2(t)=-3V_0\sin(2\omega_1t)/(8\omega_1)$. Their peak displacement <amplitudes> are $|V_0|/(2\omega_1)$ and $3|V_0|/(8\omega_1)$. Both the initial <displacement> and the initial <velocity> can be checked directly from this expression.
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