Past exam of the mathematical methods II course of the University of Oxford / 2022 / 12 / d / Solution

Substitute the three nonzero initial-velocity coefficients into the general Fourier sine series, with ${\omega }_1=\pi c/L$: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.