 Solution

Multiply the transformed equation by $\sin (k_{n}x)$ and integrate from zero to $L$. Two integrations by parts, using the zero endpoint values of both $\tilde{y}$ and the sine, giveThe delta source contributes $\tilde{F}(\omega )\sin (k_n{x}_0)$. Multiplying by $2/L$ and using the coefficient definition therefore yieldsAway from the resonant frequencies this givesAt $\omega =\pm k_n$ the quotient must be interpreted with a pole prescription if the source excites that mode. In distributional form, arbitrary delta terms at those frequencies may also be added; they are the free homogeneous oscillations. This is the frequency-space issue addressed explicitly in part (iii).