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, $c=p(0)$.
Best Answer
The equation $$p'(t)=f(t)\ p(t)$$ says: The rate $p'$ by which the population $P$ grows is proportional to the current size $p$ of $P$; but the proportionality factor $f$ valid at time $t$ depends on time. If, e.g., the growth rate depends on the seasons, the function $t\mapsto f(t)$ would be a certain periodic function with period 365 days. At any rate, the size of $P$ at some later time $t$ will be proportional to the initial size $p(0)$.
The equation $$p'(t)=\alpha p(t)+h(t)$$ with constant $\alpha\in{\mathbb R}$ says: The population $P$ would rise (or decline) at the constant rate $\alpha$, and therefore would be given by $p(t)=p(0)\ e^{\alpha t}$, if it were not for an additional extraneous influx (or decrease) $h(t)$ dependent only on time $t$, but not on the current size $p(t)$ of the population. In this case it may very well be that for large $t>0$ the initial value $p(0)$ has almost no effect on the actual value $p(t)$. This is the case, e.g., if $\alpha<0$, and $h$ is some periodic function with period $T$. Then for large $t$ there will be a certain "stable" periodic behavior. This "limiting" behavior depends only on $h$, but not on the initial value $p(0)$.
Concerning your last question: When you write the solution in the "more correct" form $$p(t)=\int_0^t e^{\alpha(t-\tau)}\ h(\tau)\ d\tau + p(0)e^{\alpha t}$$ then you can verify by inspection that $p(0)$ plays no rôle for large $t$ when $\alpha<0$.