= Solution
Use a <self-financing portfolio> which holds $\Delta=V_S$ shares and $V-SV_S$ in the <bank account>, withdrawing the claim's cash flow $C\,dt$. A share's total gain includes the <continuous dividend yield>, so its gain is $dS+DS\,dt$. Matching the claim's total gain gives
$$
dV+C\,dt=V_S(dS+DS\,dt)+r(V-SV_S)\,dt.
$$
By the <Itô formula>,
$$
dV=\left[V_t+(\mu-D)SV_S+\frac12\sigma^2S^2V_{SS}\right]dt+\sigma SV_S\,dW.
$$
The <delta hedge> cancels the <Brownian motion> term. Comparing the remaining drifts yields the <Black-Scholes equation with claim dividends>, with the stock's own dividend yield included:
$$
\boxed{V_t+\frac12\sigma^2S^2V_{SS}+(r-D)SV_S-rV=-C(S,t),\qquad V(S,T)=S}.
$$
This gain calculation accounts for rebalancing through the <self-financing portfolio> condition. The physical expected-return parameter $\mu$ cancels, as required by <risk-neutral pricing>.
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