Solving the Black-Scholes PDE via the heat equation

Starting from the Black-Scholes PDE $V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0$ for a European call with strike $K$ and expiry $T$, use a change of variables that turns it into the one-dimensional heat equation, solve that, and transform back to obtain the closed-form call price.

$C=S\,N(d_1)-Ke^{-r(T-t)}N(d_2)$, with $d_{1,2}=\frac{\ln(S/K)+(r\pm\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}$. Put $S=Ke^x$, $\tau=\tfrac12\sigma^2(T-t)$ and $V=Ke^{\alpha x+\beta\tau}u(x,\tau)$ with $\alpha,\beta$ chosen to kill the first-order and zeroth-order terms; then $u_\tau=u_{xx}$. Convolving the transformed payoff with the Gaussian heat kernel and undoing the substitutions yields the two normal-CDF terms.