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.
• So if I had something like $p'(t)=g(t)p(t)+h(t)$, g(t) might be my reproductive rate, and g(t) might be my death rate? – Korgan Rivera Sep 9 '12 at 17:37
• $\alpha$ could be the birth rate minus the death rate, and $g (t)$ could be immigration minus emigration. If more people are born than those who die, i.e., $\alpha > 0$, we have a demographic explosion. If more people immigrate than emigrate, then $g (t) > 0$. – Rod Carvalho Sep 9 '12 at 17:51
If $p'(t) = g(t)p(t)+h(t)$ and $g(t) = k$ for some constant $k$, then this form is also the simplest case of feedback control of a linear time invariant system with a state term, $p(t)$, and a control term, $h(t)$.
If $g(t)$ is not constant, then you have a controlled system where the system parameters change with respect to time.
In the time invariant case, you might describe an elevator control with a single state ($z$-position) and a single control term (motor speed). In the time varying case, then you have something directly affecting the $z$-position, for instance if the elevator isn't fixed to the cable, but starts sliding down the shaft with some velocity $g'(t)$.
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.