What is a cool topic regarding differential equations that you would suggest for self-studying? I took a basic course in differential equations and I loved it. I'd like to study them more in depth. However, I'm taking another, more advanced course in a while. For this reason, instead of self-studying what will be surely covered in class, I would like to try and read about slightly more advanced topics, to get a grasp of what cool applications of the theory or more specifics areas of study look like. Is there any topic about DEs that caught your attention, struck you as beautiful and, ultimately, would be relatively fun for an undergraduate to try and tackle, if even a little, just to see if they're really my thing?
 A: How basic is your basic Differential Equations course? Because without any doubt in my honest opinion, the Laplace transform would be an excellent choice of self study.
For two reasons I can come up with:
1) You can use the Laplace Transform to solve differential equations (in a sophisticated manner) That is the reason why the Laplace Transform usually is introduced in a DFQ course.
2) Because the theory of Laplace Transform is beautifully set up. Once you understand how it works, you can also calculate, for example, certain convergent improper integrals that would be practically impossible with an introductory calculus course where these types of integrals are introduced.
When I learned about this marvelous piece of mathematical theory, I had a hard time falling asleep that night. And yes, I consider Laplace as a greater mathematician than he usually is regarded by many, but that is my opinion.
In case you already studied the Transform (to an advanced level), then perhaps partial differential equations would be an option. That would be a next more indepth course in DFQ's
A: There's some really cool applications of imaginary numbers in differential equations: fluid modelling, Bernoulli's equation for fluids, and heat flow, I'm sure many more.  Also the proof for existence and uniqueness of 1st order ODE is fairly straightforward and interesting.  The proof for 2nd order and above is a similar principle but much more complicated. Also, it's really neat to prove that the solution space for a nth order differential equation is of space n (it's obvious, but proving it is really quite fascinating).
