# Miles Fore

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Mentor: Don Tucker
Major: Mathematics

Summer 2008 project description:

Differential equations is a broad subject, but tends to be very useful in terms of real world applications. This is because we live in a dynamic world. Things change. Change is something that can be described using differential equations. Thus it is a very important subject to study if one wants to be able to predict the future state of a system, or analyze a changing system.

The project that I am proposing for this summer mainly concerns differential equations. My past experience with this subject has been a superficial one, with the mechanics of the subject being taught rather than the underlying theory. The overall goal will be to gain a deeper understanding of differential equations and along the way (time permitting) examine some applications.

Some general topics that will be encountered along the way are non-linear ordinary differential equations, boundary value problems, and partial differential equations. Since this is such a wide range of subjects Professor Tucker will guide me as to what material I should focus on. If I am able to get far enough in the subject then one application that I would like to examine is agility. What does it mean for something to be agile? Whatever it is it seems to involve how fast something is changing, hence there should be a differential equation involved. One approach would be to analyze the path it takes- in the two dimensional case differentiate the angle the object make with the x-axis with respect to the arc length. Apparently this is the sort of thing that people who study differential geometry do, so that subject may also be encountered in the research.

The main plan of attack for this project is to begin by working through Professor Tucker's notes on differential equations. These notes tackle the subject on a deeper level then is usually taught in differential equations classes, and thus will aid in me accomplishing this project's goal. I will also (luckily) have the help of Professor Tucker to guide me and help me when I get stuck.

Summer 2008 final report