April 9 200000008

MAIN POINTS
A differential equation describes the change of some variable with respect to another. Because they are often difficult or impossible to solve exactly, they are a good candidate for numerical methods. This chapter aims to find approximations at certain points, rather than the entire solution. The text defines the "Lipschitz condition," whos significance I don't immediately see. The further theorems arising that deal with convex and Lipshitz conditions confuse me.
Euler's Method is said to be seldom used in practice. It gives an approximation to a well-posd initial value problem. The method is derived through Taylor's theorem and is a fairly simple algorithm that uses recursion. The error is shown to grow as we get away from the initial value (typical of things that use a Taylor-style of thinking). A lot of this reading was lost on me because the reading defines theorem's without first showing why they are important, and sometimes without sufficiently defining terms.

CHALLENGES
The concept of "convex" works for me, but I have a hard time imagining what the "set" looks like. Pages 251-253 really confuse me. I still don't have a very good grasp of how a differential equation works. I will definitely need to go over a lot of these things in class, and I predict other people in the class will feel the same. Euler's method seems fairly simple, though.

REFLECTION
I've never done anything with differential equations. I'm not even really sure what we'd be approximating by using Euler's method— could we be approximating the location of a pendulum, for instance? In the Euler's method we see again the use of Taylor polynomials to derive a theorem. Most of what we've used in the class has been either polynomial interpolation or using Taylor... are there any other schools of thought in approximating functions like this?