I run into a first order ordinary differential equation and am looking for various methods to solve it $$\frac{d}{dt}q(t) = aq(t)+h(t)\,g(q(t))-1,$$ where $a$ is a constant, $h$ and $g$ are two functions without knowing the explicit form.
Does this kind of differential equation has a name? Any ideas or reference would be appreciated.
As far as I know it doesn't have a special name. If $g$ happens to be a polynomial of degree $\le 3$ it's an Abel equation of the first kind. And there is no general method to solve it in closed form. For example, the differential equation
$$ y' = y + t y^3 - 1$$
probably does not have close-form solutions. Maple does not find any, nor does Wolfram Alpha.