= Solution
{c}
Let $\tau=T-t$. The <product rule> and <Itô's formula> give
$$
\begin{aligned}
d\bigl(e^{r(T-t)}V(S_t,t)\bigr)
&=e^{r\tau}\left[V_t+\frac12\sigma^2S_t^2V_{SS}+rS_tV_S-rV\right]dt
+e^{r\tau}\sigma S_tV_S\,dW_t\\
&=e^{r\tau}\sigma S_tV_S\,dW_t,
\end{aligned}
$$
where the <Black-Scholes equation> cancels the drift. Under the usual pricing growth and integrability conditions this <local martingale> is a <martingale>. Its terminal value is $f(S_T)$, since the exponential factor becomes one at $T$. Taking a <conditional expectation> gives
$$
e^{r(T-t)}V(S_t,t)=\mathbb E[f(S_T)\mid\mathcal F_t],\qquad
\boxed{V(S,t)=e^{-r(T-t)}\mathbb E[f(S_T)\mid S_t=S].}
$$
The dynamics already use the <risk-neutral measure>. This is the <Feynman-Kac formula> for the given pricing problem; a differential equation alone, without the relevant growth condition, would not justify upgrading a local martingale or uniqueness.
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