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.
Best Answer
My real advice is to start with something less mathematical and more intuitive so that you understand the motivations for the more mathematical approaches.
That having been said, if you're looking for a mathematically rigorous approach to ideas of central importance in economic theory, I'd start with Debreu's Theory of Value. Mas-Colell (suggested in another answer) is a massive textbook that touches on a gazillion different topics; Debreu reads like a math paper. It's dated, but only in the same sense that, say, Serre's FAC is dated. It's a seminal work, it's the inspiration for a lot of what's come since, and you can still learn a lot from it.