= Solution
Differentiating the <European call option> price in strike gives, for $T>0$,
$$
C_K(T,K)=-e^{-rT}\int_K^\infty\psi(T,s)\,ds,
\qquad C_{KK}(T,K)=e^{-rT}\psi(T,K).
$$
The strike derivative uses dominated convergence; the second uses the continuous density. Also,
$$
\int_K^\infty s\psi(T,s)\,ds
=e^{rT}\bigl(C(T,K)-KC_K(T,K)\bigr).
$$
Differentiate the supplied time-integral identity and the discount factor. Continuity of the density supplies the diffusion-term derivative. For the tail first moment, continuity in time follows from continuous <stock> paths, locally uniformly bounded second moments, and the absence of an atom at $K$. Therefore
$$
\begin{aligned}
C_T&=-rC+e^{-rT}\left(r\int_K^\infty s\psi(T,s)\,ds
+\frac12K^2\sigma(T,K)^2\psi(T,K)\right)\\
&=-rKC_K+\frac12K^2\sigma(T,K)^2C_{KK}.
\end{aligned}
$$
Hence the <Dupire equation> is
$$
\boxed{C_T(T,K)=\frac12K^2\sigma(T,K)^2C_{KK}(T,K)-rKC_K(T,K).}
$$
Its initial condition is $C(0,K)=(S_0-K)^+$; natural strike boundaries are $C(T,0)=S_0$ and $C(T,K)\to0$ as $K\to\infty$. These are consistent with the discounted <stock> <martingale> and integrable tails. Where $C_{KK}>0$, the same identity gives <local volatility recovery from call prices>:
$$
\boxed{\sigma(T,K)^2=
\frac{2(C_T+rKC_K)}{K^2C_{KK}}.}
$$
The equation evolves in maturity and strike, unlike the backward option-value equation in calendar time and spot.
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