This chapter contains five projects which develop the ideas and techniques introduced in this book. The first four projects are all based on recent research publications. Their purpose is to illustrate the variety of ways in which ODEs arise in contemporary research - ranging from engineering to differential geometry - and to provide an authentic opportunity for the reader to apply the techniques of the previous chapters. If possible, the projects could be tackled by a small group of students working as a team. The fifth project has a different flavour. Its purpose is to guide the reader through the proof of the Picard-Lindelöf Theorem. At the beginning of each project, I indicate the parts of the book which contain relevant background material.

Ants on polygons

(Background: Chapters 1 and 2, Exercise 2. 6)

Do you remember the problem of four ants chasing each other at constant speed, studied in Exercise 2. 6? We now look at two variations of this problem. In the first, we consider n ants, where n = 2, 3, 4, 5 …, starting off on a regular n-gon. Here, a 2-gon is simply a line, a regular 3-gon an equilateral triangle, a 4-gon a square and so on. In the second, more diffcult, variation we consider 4 ants starting their pursuit on a rectangle with side lengths in the ratio 1:2. This innocent-sounding generalisation turns out to be remarkably subtle and rich, and is the subject of recent research reported in Chapman, Lottes and Trefethen [4].