It sounds like you also want an introduction to differential geometry, as well as a good grounding in ODE's. As an undergraduate, I had Martin Braun's book on differential equations and their applications, and Barrett O'Neill's Elementary Differential Geometry. They should be quite approachable, and a thorough reading should give you enough background for later courses.

I recommend doing some of the computations, because knowing some of the numerical analysis issues can be important, even though they are addressed less than superficially if at all in these books. Also, it's important that you "feel you could start doing calculus on a Moebius strip", at least locally, even if you don't actually do it. Such a feeling can give one comfort when one approaches the subject in depth. I missed having such a bedrock in some of my analysis/PDE courses; this may be why I ended up doing more algebra and logic instead.

Gerhard "Ask Me About System Design" Paseman, 2010.06.18