### MA 532-001 Fall 2012 Ordinary Differential Equations I

Mathematics

Ordinary Differential Equations by Schaeffer and Cain | Feedback on text
Ordinary Differential Equations and Dynamical Systems by Gerald Teschl
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1. Schaeffer and Cain, Chapter 1, Introduction

• Phase portrait of x'=f(x), x in R (the "phase line"): Teschl book, p. 19 to top of p. 20.
• Phase portrait of x''+f(x)=0 (section 12.4 of Nagle, Saff, and Snider, Fundamentals of Differential Equations and Boundary Value Problems, 5th edition, 2008). The idea here is similar to that described in lecture, but where they graph a function they call G(x), we graphed it's opposite -G(x).
• Euler-Lagrange equations (much more than we did; source is http://www.people.fas.harvard.edu/~djmorin/book.html)

2. Schaeffer and Cain, Chapter 2, Linear systems

• General theory for x'=A(t)x: Teschl book, section 3.4.
• x'=Ax: Teschl book, sections 3.1-3.2.
• More on computing Jordan form: Schaeffer and Cain, Appendix C (not yet included in the pdf file of the text).

3. Schaeffer and Cain, Chapter 3, Nonlinear systems: local theory

4. Schaeffer and Cain, Chapter 4, Nonlinear systems: global theory

### Resources

• Newton's anagram

• dfield and pplane

• OdeFactory. I just learned about this program, which does much more that dfield and pplane, and is more pleasant to use and look at than xppaut. At the link above, click on the second menu item, How to Get OdeFactory, to learn how to install this program on your computer.

• Xppaut

To get started, go here and click on Software Installation for instructions.

Once you have xppaut working, go back here, click on Chapter 1, and download four files:

ch-1-riccati.ode
ch-1-riccati.pdf
ch1-van-der-Pol.ode
ch1-van-der-Pol.pdf

The .ode files will run on xppaut. The .pdf files are instructions for what to do with them. These instructions will teach you a little about xppaut.

Xppaut is based on a program that was originally written for Windows, so it should work well on a Windows machine. If you use Linux, you are probably knowledgeable enough to deal with any issues that arise.

The following is for mac users.

Following all the instructions will take you to here. Then click on "latest", which will take you here. (Or just go there directly.) Download the disk image xpp611fosx.dmg. Open it to produce a volume that contains the folder xppmac. Put this folder in some convenient place, like your Desktop. Open the folder. Drag the file xppaut to your Applications folder, and drag the icon xpp to some convenient place, like your Dock. To test the program, open the folder ode and drag one of the .ode files onto the xpp icon. xpp should start.

Warnings for mac users:

(1) You must be using os 10.6 or greater. There are instructions for 10.5, but they have not worked for me.

(2) You must already have an X server installed. This is a program in your Applications/Utilities folder named X11 or something similar. If it's not there, it may be on a cd that came with your mac. Alternatively you can search the web for a free X11 server for the mac, such as XQuartz.

(3) I've encountered a few idiosyncracies on a mac. Clicking on the little red circle that should make a window go away does not work. If you finish working with one .ode file and want to work on another, I think you have to quit xppaut and start it up again.