= Solution
Multiply the transformed equation by $\sin(k_nx)$ and integrate from zero to $L$. Two <integrations by parts>, using the zero endpoint values of both $\widetilde y$ and the sine, give
$$
\int_0^L\widetilde y_{xx}\sin(k_nx)dx=-k_n^2\int_0^L\widetilde y\sin(k_nx)dx.
$$
The delta source contributes $\widetilde F(\omega)\sin(k_nx_0)$. Multiplying by $2/L$ and using the coefficient definition therefore yields
$$
(k_n^2-\omega^2)b_n(\omega)=\frac2L\sin(k_nx_0)\widetilde F(\omega).
$$
Away from the resonant frequencies this gives
$$
\boxed{b_n(\omega)=\alpha_n\frac{\widetilde F(\omega)}{k_n^2-\omega^2},\qquad\alpha_n=\frac2L\sin(k_nx_0).}
$$
At $\omega=\pm k_n$ 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).
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