To paraphrase the Oldsmobile ad, "This is not your father's differential equations course." The traditional course in differential equations focused on producing formulas for solutions to those few differential equations for which analytic expressions for the solutions are possible. These formulas allowed calculation of the values of the solutions at any point, but often left open important questions like the long term behavior of the solutions. The easy availability of computers now allows us to compute numerical approximations to the solutions of a large number of differential equations, shifting the focus to more conceptual issues. The new aproach focuses on understanding the solutions directly from an analysis of the differential equation itself. The essential ideas include the study of the phase plane, linearization near equilibrium solutions, bifurcations, conserved or dissipated quantities, and effective methods of approximation. In applications of differential equations, this sort of analysis has always been essential.
Our goal will be to produce differential equations which model phenomena, to understand the nature of the solutions, and to interpret the behavior of the solutions in terms of the original phenomena. At this stage, the solutions may predict behavior consistent with the phenomena, but often we will find that we have to improve our model and repeat the process.
Two software packages will be used extensively in the course, and will be available without charge in the lab in 31 State Hall. The first is IDE, or Interactive Differential Equations, which provides over ninety illustrative demos concerning differential equations. The cost for your own copy to run at home is approximately $20 (CD only) ISBN #0-201-57139-0 or $30 (CD and workbook) ISBN 0-201-19228-4. This is available from Addison-Wesley Interactive. This software is designed for either Macs or PCs.
When you are in the lab in 31 State Hall, you can start up IDE by clicking on this .
The second package consists of MATLAB routines written by John C. Polking. In the 31 State Hall lab, start up MATLAB 5 (the default version of MATLAB: version 4 of MATLAB is available also, but we will not need it), and the routines you need are called
and may be found in the directory T:\MAT235\DIFFEQ\POLKING. (This directory should already be in the path MATLAB searches.) To use these with your home computer, you can download the latest versions from Professor Polking's web page cited above, or copy them from my copies. I do not have an anonymous ftp server running here, so you should copy mine by using Netscape's File --> Save As option on the links above, or follow the links on John Polking's page cited above. These routines are copyrighted by John C. Polking and are made available for free for educational use. Anyone else wishing to use them should contact John C. Polking.The program dfield is for one dimensional differential equations. The other, pplane, is for systems. To run either program, first open MATLAB. When the program is running, type dfield5 or pplane5 at the >> prompt. The program will open a setup window in which you enter the equation, any parameters, and the region in the plane to be graphed. When you click on the button at the lower right of the setup window labelled "Proceed" the program will then open a graphics window in which the results will be displayed.
In addition, there are many resources for differential equations available on the the Web, including Java applets to plot phase planes from
Here are some collections of information about Differential Equations you may wish to look at as well:
| 3 In-class exams | 300 |
| 4 Labs | 200 |
| Final | 200 |
| Total | 700 |
Makeup exams will not be given, since no one item contributes a commanding portion of the grade. If you have a legitimate excuse for missing an exam, it will omitted from the calculation, so that your grade will be based on a total of 600 rather than 700 points. If you do not have a legitimate excuse, your grade will be 0. In general, a legitimate excuse is one over which you do not have control and which you could not reasonably anticipate. For example, a late bus, or other exams the same day, can be anticipated, and would not be considered legitimate.
Late labs will be reluctantly accepted but will be discounted 25 percent for each day (or fraction thereof) they are late. Labs should be writen up in grammatical English. Formulas and graphs with no explanation of their meaning will not be given much credit.