What can this differential equation be used to model? So, I can model growth and decay if I start with assuming that the growth rate is constant:

$\frac{p'(t)}{p(t)}=\alpha$

and then I have 

$p'(t)-\alpha p(t)=0$

A general linear differential equation, however, would have the form

$p'(t)-g(t) p(t)=h(t)$

So the growth rate is g(t).  What is h(t)?
And what type of thing is this form of equation used to model?

This question has been edited.
You can read it like this: $p'(t)= \alpha p(t)+g(t)$. Then the growth is "caused" by "reproduction" plus "immigration", for example.
I know you asked about first order, but: Second order homogeneous ODEs, e.g. $a\ddot{x} + b\dot{x} + cx = 0$, describe "unmolested" damped/driven harmonic oscillation. Introducing a function on the right hand side, e.g. $a\ddot{x} + b\dot{x} + cx = f(t)$, describes damped/driven harmonic oscillation with $f$ describing an external input. The part of the ODE which does not involve a dependent variable describes "outside influences". I am quite confident that this is the same for any order of ODE.