Mathematics 207A, ODE
Syllabus
Fall Quarter 2012

Instructor: Jim Kelliher
Office: Surge 232 moving to Surge 224
Office Hours: Whenever my door is open, no one else has my attention, and I am not preparing for a class or something like that
Email: kelliher

First, to business:

Text: There will be no official text for this course (see below).

Homework: Homework will be assigned in class on a regular basis. You are free to work together as much as you want on the homework, but your writeup must be your own.

The exams: There will be one in-class midterm and one cumulative in-class final. We will discuss when to schedule these.

Calculating your grade: Your total numeric score for the course will be based on three scores, each normalized to be out of 100: homework, midterm, and final. The total numeric score will be the higher of the following:

• 25% of homework + 25% of midterm + 50% of final.


• 25% of homework + 10% of midterm + 65% of final.

Hence, you will have a chance to partially make up for a poor showing on your midterm. I will not disregard your homework scores, though, as the homework will contain the lengthier and more difficult problems.

Now a few words about the course.

The Math 207A, B, and C sequence is being taught for the first time here at UCR, so in this regard we are entering new territory together. (Math 207A, however, does share a title with the now defunct, though still in the course catalog, Math 211A.)

Ordinary differential equations (ODEs) are useful mathematical models of processes occurring in the real world as well as a basic tool used in establishing the existence of solutions to partial differential equations (PDEs). One point of view is that an ODE represents a continuous-time finite-dimensional dynamical system, while a PDE represents a continuous-time infinite-dimensional dynamical system. This dynamical systems point of view is the one we will mostly adopt, as it provides a framework that can encompass all of ODE theory.

I know that many of you will have never taken an ODE course as an undergraduate. And some of you who have will call those students lucky. This is because it is the ugly aspects of the subject—a hodgepodge of techniques for solving various types of ODEs—that are usually taught first to undergraduates. We will not be going down that road. This is not to say that the road we'll go down will be easier, just that the scenery will be more aesthetically pleasing.

So anyway, why is there no text for this course? It's because the only excellent ODE book I know of that adopts at least in part the dynamical systems point of view that does not assume too much background in real analysis is the undergraduate text by Boyce and DiPrima. This text also starts with the hodgepodge approach (though well done in my opinion) but adopts the dynamical systems point of view (though not explicitly) when it begins to discuss systems of ODEs.

What I intend to do is to start by covering this material in Boyce and DiPrima with somewhat more rigor and at a depth more appropriate for graduate school. And with an eye toward its utility for Math 207B. I will also discuss Sturm-Liouville boundary value problems, and some applications of ODEs to PDEs; in particular, the method of characteristics, a way of reducing first-order PDEs to a system of first-order ODEs.

Here are some texts that you might want to consult if you want to read up a little on ODEs, either before, during, or after this course:

• n-th edition of Boyce and DiPrima's, Elementary Differential Equations and Boundary Value Problems. I think n has reached 8 now, but any version will do, the cheaper the better. In the seventh edition, Chapters 7 and 9 are the key ones, along with Chapter 11 on Sturm-Liouville boundary value problems.


• Fourth Edition of Ordinary Differential Equations by Garrett Birkhoff and Gian-Carlo Rota. The first edition of the text was from 1959, and the text is still largely of that era. It has mostly the same material as Boyce and DiPrima, though with greater expectations of mathematical maturity, with more proofs and with a just slightly more abstract attitude. But with less of a dynamical systems feel to the analysis of systems of ODEs, especially as much of this is in an appendix. Still, it has complete proofs of most of the things we will talk about, even if they are a little hard to understand at times.


• Second Edition of Ordinary Differential Equations by Philip Hartman. The first edition of this text is from the same era as that of Birkhoff and Rota's, and I would make much the same comments about it. It is at a more advanced level mathematically than Birkhoff and Rota, I would say. It also covers the method of characteristics, among other topics.

There are, of course, a gazillion other books, about half of them simply called, Ordinary Differential Equations. But one can't read everything.

Last confirmed edit: 21 August 2012
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(thanks Phil)