References

Okay, I'm about to take an introductory course for PDE's (for some reason I got into weak solutions and such before learning the preliminaries), and so I would like to know if there are any books you would recommend that have a strong mathematical orientation (not that I don't like the many applications, but when I took an ODE's course we spent almost all semester modelling populations and such and when I read on my own Picard-Lindelöf and Peano's existence theorems, and all the mathematics behind what I was actually doing I felt cheated).

Well, for ODE's, I can say that Coddington and Levinson is a great resource. Every time I needed some theorem or other, such as the dependence of solutions on initial conditions, my professors would say, "Well, that's in Coddington and Levinson." The book is Theory of Ordinary Differential Equations. As for PDE's, I would think something like Renardy and Rogers might be good. That's an introductory graduate-level text that's sure to have plenty of theory. The book there is An Introduction to Partial Differential Equations.

If you like the theory of PDEs, I might suggest "Partial Differential Equations" by Evans published by the AMS.