## Ordinary Differential Equations

This book consists of ten weeks of material given as a course on ordinary differential equations (ODEs) for second year mathematics majors at the University of Bristol. It is the first course devoted solely to differential equations that these students will take.

An obvious question is: why does there need to be another textbook on ODEs? From one point of view the answer is certainly that it is not needed. The classic textbooks of Coddington and Levinson, Hale, and Hartman provide a thorough exposition of the topic and are essential references for mathematicians, scientists and engineers who encounter and must understand ODEs in the course of their research. However, these books are not ideal for use as a textbook for a student’s first exposure to ODEs beyond the basic calculus course (more on that shortly). Their depth and mathematical thoroughness often leave students that are relatively new to the topic feeling overwhelmed and grasping for the essential ideas within the topics that are covered. Of course, (probably) no one would consider using these texts for a second year course in ODEs. That's not really an issue, and there is a large market for ODE texts for second year mathematics students (and new texts continue to appear each year). I spent some time examining some of these texts (many which sell for well over a hundred dollars) and concluded that none of them really would work for the course that I wanted to deliver. So, I decided to write my own notes, which have turned into this small book. I have taught this course for three years now. There are typically about 160 students in the class, in their second year, and I have been somewhat surprised, and pleased, by how the course has been received by the students. So now I will explain a bit about my rationale, requirements and goals for the course.

In the UK students come to University to study mathematics with a good background in calculus and linear algebra. Many have seen some basic ODEs already. In their first year students have a year long course in calculus where they encounter the typical first order ODEs, second order linear constant coefficient ODEs, and two dimensional first order linear matrix ODEs. This material tends to form a substantial part of the traditional second year course in ODEs and since I can consider the material as already seen, at least once, it allows me to develop the course in a way that makes contact with more contemporary concepts in ODEs and to touch on a variety of research issues. This is very good for our program since many students will do substantial projects that approach research level and require varying amounts of knowledge of ODEs.

This book consists of 10 chapters, and the course is 12 weeks long. Each chapter is covered in a week, and in the remaining two weeks I summarize the entire course, answer lots of questions, and prepare the students for the exam. I do not cover the material in the appendices in the lectures. Some of it is basic material that the students have already seen that I include for completeness and other topics are "tasters" for more advanced material that students will encounter in later courses or in their project work. Students are very curious about the notion of chaos, and I have included some material in an appendix on that concept. The focus in that appendix is only to connect it with ideas that have been developed in this course related to ODEs and to prepare them for more advanced courses in dynamical systems and ergodic theory that are available in their third and fourth years.

There is a significant transition from first to second year mathematics at Bristol. For example, the first year course in calculus teaches a large number of techniques for performing analytical computations, e.g. the usual set of tools for computing derivatives and integrals of functions of one, and more variables. Armed with a large set of computational skills, the second year makes the transition to `'thinking about mathematics'' and creating mathematics. The course in ODEs is ideal for making this transition. It is a course in ordinary differential `'equations'', and equations are what mathematicians learn how to solve. It follows then that students take the course with the expectation of learning how to solve ODEs. Therefore it is a bit disconcerting when I tell them that it is likely that almost all of the ODEs that they encounter throughout their career as a mathematician will not have analytical solutions. Moreover, even if they do have analytical solutions the complexity of the analytical solutions, even for `'simple'' ODEs, is not likely to yield much insight into the nature of the behavior of the solutions of ODEs. This last statement provides the entry into the nature of the course, which is based on the vision of Poincare--rather than seeking to find specific solutions of ODEs, we seek to understand how all possible solutions are related in their behavior in the geometrical setting of phase space. In other words, this course has been designed to be a beginning course in ODEs from the dynamical systems point of view.

I am grateful to all of the students who have taken this course over the past three years. Teaching the course was a very rewarding experience for me and I very much enjoyed discussing this material with them in weekly office hours.

This book was typeset with the Tufte latex package. I am grateful to Edward R. Tufte for realizing his influential design ideas in this Latex book package.