Solution
= Solution
For $k\ne0$, the <chain rule> gives $g_t=-f_tg/(2k)$ and $g_{xx}=(-f_{xx}/(2k)+f_x^2/(4k^2))g$. Consequently
$$
g_t-kg_{xx}=-\frac g{2k}\left(f_t+\frac12f_x^2-kf_{xx}\right)=0.
$$
Thus
$$
\boxed{g_t=kg_{xx}.}
$$
This is the potential form of the <Cole-Hopf transformation>, with $y=-2k\,g_x/g$. For $k>0$ it is a forward <diffusion equation>; at $k=0$ the exponential transformation is undefined, while negative $k$ gives a backward heat equation.