Suppose you maintain a pond with fish (for profit, of course, this is economics!).
When the food is abundant and there are not many fish, the population grows at a constant rate
$k>1$ (reproduction rate minus death rate), so we have $y'=ky$. This is separable. Solve it. Give a numerical example.
Conclude from the example that our assumptions are not realistic. So what is wrong with our assumptions?
Abundant food!!!
(Of course. This is economics after all:-)
The next simple assumption is that the pond can support only some maximal population, say $A$.
Which means that when the population approaches $A$ the death rate increases (starvation), so the net
growth rate is not just $k$ but $k(1-y/A)$. When $y$ is small, (or $A$ is very large) we
have almost $y'=ky$ as before. When $y$ is close to $A$, the net rate of change approaches $0$,
as it should be. We obtain $y'=ky(1-y/A)$, another separable equation!
But this pond brings you no profit yet. To make a profit, you have to catch some fish, say at
a constant rate. You obtain another separable equation $y'=ky(1-y/A)-c$. Discuss what happens
for various values of parameters $k,A,c$.
And so on:-)
You can go further and further with this model when time permits. Suppose that
instead of harvesting a fixed amount $c$, you gauge the population somehow, and harvest $cy$, a fixed proportion of the population. This leads to another separable equation, as well as to a useful discussion, which strategy is better, $c$ or $cy$ in terms of long term profits and in terns the pond sustainability.
Then, if time permits, you can pass to two functions and systems of equations.
The classical example is Volterra-Lottka system, which involves a slightly more complicated
ODE, but it is also separable. And its original motivation was also economics: the influence of World War I on the population
of sardines in the Mediterranean (an important economic resource for surrounding countries).
Remark. Besides fish, there are somewhat similar models of warfare (also a kind of economics btw), search on "Lanchester laws"; they lead to simple 2x2 systems of linear differential equations, and they have been compared to what happens in real wars.
