Consider a function $h(x, t) = \exp(t\cdot f(x) + g(x))$ given, and $\alpha(x, t)$ unknown. Can we say anything about the following non-homogenous ODE:
$\partial_t h = -h \cdot \partial_x \alpha - \alpha \cdot \partial_x h$,
with $\alpha: \mathbb{R} \times [0,1] \rightarrow \mathbb{R}$ (the time requirement can be relaxed.)
Let's forget about $t$ as a variable, and simply rewrite the equation as:
$$(\alpha(x) h(x))'+\alpha(x) h(x)'+f(x) h(x)=0$$
Expanding, we have:
$$h \alpha'+2h' \alpha+fh=0$$
We will also rename the function (again, treating $t$ as a constant parameter):
$$s(x)=t f(x)+g(x) \\ h=e^s \\ h'=s' h$$
We now have:
$$\alpha'+2s' \alpha+f=0$$
This is a simple 1st order linear ODE, which can be solved by usual methods.
Then we remember that:
$$s'=t f'+g'$$
So we also have $\alpha$ depend on $t$.