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 assumtions?
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:-)
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. 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).
