(Last modified Dec 13, 2002.)
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.
The MATLAB routines we will use to solve differential equations were written by John C. Polking. Their names are dfield and pplane, with a number appended, corresponding to the version of MATLAB for which they are designed. The instructions below refer to dfield5 and pplane5, since that was the version I was using when they were written. I sometimes use dfield6 and pplane6 since they incorporate some inprovements over dfield5 and pplane5. Note that you should also retrieve the program ppnout, as it handles output of the results. You may retrieve any of several versions from the dfield web page . These routines are copyrighted by John C. Polking and are made available for free for educational use.
The program dfield is for one dimensional differential equations. The other, pplane, is for 2 dimensional systems. To run either program, first open MATLAB. When the program is running, type dfield{ 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 a Java applet to plot phase planes from
Here are some collections of information about Differential Equations you may wish to look at as well:
First, READ THE BOOK . You should read each section before we talk about it in class, then again after class, before doing the homework for the section. If you have any trouble understanding it, read it several times, first, quickly for an overall idea what the section is about, then in detail, working out the examples the book uses to make sure you know why each statement is true. Only after this should you start the homework. You may be pleasantly surprised how much easier the homework is with this sort of preparation. You will certainly understand the material and retain more of it, if you study in this way.
Your grade will be determined by your scores on 2 in-class tests, worth 100 points each, on 3 labs, worth 50 points each, and a comprehensive final exam, worth 200 points, for a total of 550 points possible.
| 2 In-class exams | 200 |
| 3 Labs | 150 |
| Final | 200 |
| Total | 550 |
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 500 rather than 600 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.