I'm a business operations guy. I've come to really enjoy reading up on the math, stats, and coding that comes with the models/tools that we use. I specify this so that you have a practical frame of reference: I'm not a physics undergrad student who needs to solve ODEs and/or PDEs by hand for class. Rather, I'm interested in (A) Numerical approximations via python to ODEs/PDEs and (B) examples of the sort of problems they can solve.
What originally perked my interest was usage of Hamiltonian Monte Carlo based samplers, which use differential equations to inform a dynamic path across probability distributions. (HMC is implemented in PyMC3, so I don't need to implement this myself.) Anyway, I did further reading and learned that differential equations are also used heavily in ecology and economics.
The potential applications seem limitless and so I'd like to get some exposure to A & B (above.) Any recommendations on books and or courses would be appreciated!
You might enjoy CFD Python: 12 steps to Navier-Stokes. CFD is computational fluid dynamics, and Navier-Stokes are the partial differential equations that govern fluid flow in a variety of conditions.
The best part of CFD is all the pretty visualizations you will make :)