If I have this equation:
$p'(t)-p(t)\alpha =0$
I can say that $p$ is a function that represents the size of a population at time t. The rate at which the population grows is constant. The solution will show that the size of the population is proportional to the initial size.
If I have this equation:
$p'(t)-p(t)f(t) =0$
I can say the rate at which the population grows is determined by $f$. The size of the population is still proportional to the initial size.
But if I have:
$p'(t)-p(t)\alpha = h(t)$
It's difficult to determine from the solution what role the initial size of the population, $p(0)$, has. The solution is:
$p(t)=\bigl(\int e^{-\alpha t}h(t)\ dt + c\bigr)\ e^{\alpha t}$
So my question is this: if I'm interpreting these differential equations as growth functions, what does $c$ represent in the last equation? In the previous equations, $c=p(0)$.