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.