I have already taken an ODE course and I passed it well but I learned nothing after passing it. I mean I can solve problems by applying the techniques I have been taught on them, but there are still a lot of things I don't understand such as:
Why does Fourier series gain so much attention when it's possible to write other orthonormal expansions as well?
I like to learn more about Laplace transformations and other types of transformations that my course did not cover (such as Z-transformation, for example)
I like to learn more about special functions that arise in physical problems.
I like to learn about famous problems in physics, mathematics and engineering that opened the door for new insights in mathematics.
I like to learn about PDEs without going into too much detail like functional analysis, etc.
Does every ODE/PDE come with a natural choice of orthonormal expansion? Or what is the relation between a differential equation and expressing the solution as a series? (Do I make sense here? Because I suspect that there must be some sort of relation, but I can't formulate it or find it on my own)
Links between ODEs/PDEs and complex analysis
In short, I'm looking for a book that emphasizes important concepts behind differential equations and has a lot of good problems with applications to physics and engineering too. If the book comes with a manual, then that would be perfect. Any suggestions?
Edit: It doesn't have to be just one book. If you can suggest two books that complement each other and answer my question, that's fine.