Table of Contents

• Class data
• Professor
• Schedule
• Introduction
• Requirements and grading
• Assignments
• MATLAB, etc.

Class:

• Math 2150, Section 003
  
• 325 State Hall
  
• MW    5:30 - 7:20
• Texts:
  • Fundamentals of Differential Equations, 5th ed.,Nagle and Saff
  • Matrix Operations, Bronson

Professor:

• Robert Bruner
  
• Office: 1201 FAB,   (313)-577-3179
• Math Department:   (313)-577-2479
• Fax:   (313)-577-7596 (shared)
• e-mail: rrb@math.wayne.edu
• Office hours: Tue + Wed 11:45 - 1:00 (1201 FAB) and Wed 7:20 - 8:00 (325 State).
  Short questions can often be answered before or after class. I will always be available in the scheduled location during my office hours. In addition, you are welcome to make appointments to see me at other times. If you are willing to take a chance that I am busy or not in, you can simply drop by my office. Finally, email is often a very effective means of getting in touch with me.

Schedule:

Here is a schedule of the topics and sections we will cover, as well as the timing of the tests.
F = Fundamentals of Differential Equations,
M = Matrix Operations

Introduction	F, Chap. 1 and 2	2 weeks
Modelling	F, Chap. 3	1 week
Linear Algebra	M, Chap. 1-6	2 and 1/2 weeks
Test 1	Wed. Oct. 10	(subject to change)
Higher order equations	F, Chap. 4 and 6	3 weeks
Test 2	Wed. Nov 14	(subject to change)
Laplace Transforms	F, Chap. 7	2 weeks
Power series	F, Chap. 8	1/2 weeks
Thanksgiving Break	Nov 22 and 23
Linear Algebra (cont.)	M, Chap. 7	1 week
Systems	F, Chap. 9	1 and 1/2 weeks
Final Exam	Mon, Dec 17	5:30 - 7:20 PM, 325 State Hall

Introduction

This is a payoff course, where everything you have learned up to this point will come together to allow you to solve much more sophisticated problems.

Linear equations can be solved quickly and efficiently, and lead to notions like the number of degrees of freedom of a system, or the number of independent constraints on a system.

In algebra and in much of calculus, solutions to problems consist of single numbers: the maximum volume, or minimum perimeter, or an area or speed. In contrast, the solution to a differential equation is a function which might express the trajectory of an object or the response of a system to a varying input. An enormous amount of insight into the behavior of systems can be gained from studying the differential equations these systems obey.

The two subjects interact strongly because the differential equations we understand best are the linear ones, and tools from linear algebra help in their solution, while their solutions display the meaning of some of the ideas from linear algebra. Nonlinear problems are often studied by their linear approximations, so that understanding the linear equations is a key step in the analysis of all differential equations.

Requirements and grades:

First, READ THE BOOKS . 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.

Second, regular attendance and class participation will be expected. The only way to learn mathematics is to do it. You should make sure you understand the homework assigned. I will answer questions about it in class, before or after class, or in office hours.

There will be three 'labs', or extended problems, which require you to analyze a differential equation or family of equations, and reach both qualitative and quantitative conclusions. Think of yourself as a design engineer or as a consultant brought in to solve a problem and provide a clear and concise solution. You will be taught the computer programs needed in these projects, and will have a week or two to work on them (varying with their complexity).

Grades will be computed as follows:

Software:

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 now 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 , or copy version 6 (for MATLAB 6.0) from

• dfield6.m
• pplane6.m
• ppn6out.m

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 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 a Java applet to plot phase planes from

• Richard Mansfield at Penn State.

Here are some collections of information about Differential Equations you may wish to look at as well:

• The Boston University Differential Equations Project
• The Consortium for Ordinary Differential Equations Experiments

Assignments