Transform heat equation to add drift/transport term

Question

Let $f \in C^{1,2}((0,\infty)\times \mathbb R)$ be a solution to the heat equation: $$ \partial_t f(t,x) =\partial_x^2f(t,x). $$ Given a constant $c\in \mathbb R$, is there a reasonable transformation $f \mapsto g$ such that $g\in C^{1,2}((0,\infty) \times \mathbb R)$ solves $$ \partial_t g(t,x) = c ~\partial_x g(t,x) + \partial_x^2 g(t,x)\quad ?$$

Answer 1

The solution consists in taking $g(t,x) = \exp(at+bx) f(t,x)$ for reasonably chosen $a,b\in\mathbb R$. Namely, $a = -2c$, $b = c^2$.