= Solution
For <claim dividends proportional to the underlying stock>, put $V(S,t)=S\phi(t)$. Then $V_S=\phi$, $V_{SS}=0$ and $V_t=S\phi'$, so the <partial differential equation> reduces to the <linear ordinary differential equation>
$$
\phi'-D\phi=-f(t),\qquad\phi(T)=1.
$$
The <integrating factor> $e^{-Dt}$ gives $(e^{-Dt}\phi)'=-e^{-Dt}f(t)$. Integrating backwards from the terminal value,
$$
\boxed{V(S,t)=S\left[e^{-D(T-t)}+\int_t^T e^{-D(u-t)}f(u)\,du\right]}.
$$
The formula assumes the displayed <integral> is finite. Under the <risk-neutral measure>, $\mathbb E_t[S_u]=S e^{(r-D)(u-t)}$. Discounting each cash payment at its own payment time gives the same result:
$$
V=\mathbb E_t\!\left[e^{-r(T-t)}S_T+\int_t^T e^{-r(u-t)}f(u)S_u\,du\right].
$$
The <interest rate> disappears because every payment is proportional to the <stock> price.
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