Instructor
Required Text
Differential Equations by Paul Blanchard, Robert Devaney and Glen Hall
Computer Software
WinPlot
Maple V, Release 6
Maple is a powerful computer algebra system that is available over the network in all computer labs. WinPlot is a free program that you can copy to your own computer.
Supplemental Materials
The following items are on reserve in the Library:
Student Solution Manual to the textbook
Martin Braun, Differential Equations and Their Applications
Robert Borrelli and Courtney Coleman, Differential Equations, A Modeling Perspective
Borrelli, Coleman, and Boyce, Differential Equations Laboratory Workbook
Goals
• Learn to recognize special classes of differential equations and understand how to solve them
• Be able to use computer graphics to analyze the behavior of differential equations
• Understand and make use of differential equations in modeling
• Be aware of some of the theory regarding solutions of differential equations
• Learn to communicate mathematics effectively, both verbally and in writing.
Topics
• Chapter 1 (First Order Differential Equations)—Sections 1–5, 8, Appendix A
• Chapter 2 (First Order Systems)—Sections 1–3
• Chapter 3 (Linear Systems)—Sections 1–4, 6–8
• Chapter 5 (Nonlinear Systems)—Sections 1, 2, 5
• Chapter 4 (Forcing and Resonance)—Sections 1–3
In this course we will study solutions of ordinary differential equations using a three-pronged approach. Solutions are obtained using analytic, geometric, and numerical techniques. All three approaches have their advantages, and we will learn when to use the appropriate technique. We will begin by deriving a few classical examples with an emphasis on the phenomena that they model. We will then discuss first-order equations using all of the techniques mentioned above. Next we will study first-order systems. Using techniques from linear algebra, we will derive a systematic approach to the solutions of linear systems. Unfortunately, nonlinear systems are more difficult to investigate, but we will learn how to apply what we know from the linear case to the nonlinear case. While we will not cover every section in the text, they are all interesting and important, and I encourage you to read some of the sections we skip.
You may need to review some calculus material, particularly some of the basic integration techniques (substitution and integration by parts) and Taylor series.
Office Hours
I do not have regularly scheduled office hours. Rather, you are encouraged to stop by my office at any time when you have questions or problems and if I am not too busy I will be happy to work with you. You may also stop by to make an appointment for a time that is mutually convenient. Another good way to contact me is through email, particularly during the evenings or weekends. I promise to respond to your email as quickly as I can.
Computing Resources
We will make frequent and important use of computer technology to help us learn about differential equations. The program WinPlot will be used for the graphical exploration of solutions to differential equations. Analytic and numeric solutions to complement our graphical work will be obtained with the help of the computer algebra system Maple. The philosophy we will take can be summarized by the following quote from Elementary Differential Equations, 5th Edition, by William Boyce and Richard DiPrima:
"For you, the student, these various computing resources have an effect on how you should study differential equations. It is still essential to understand how the various solution methods work, and this understanding is achieved, in part, by working out a sufficient number of examples in detail. However, eventually you should plan to delegate as many as possible of the routine (often repetitive) details to a computer while you focus more attention on the proper formulation of the numeric, graphic, and analytic methods so as to attain maximum understanding of the behavior of the solution and of the underlying process that the problem models. Our viewpoint is that you should always try to use the best tools available for each task. Sometimes this is a pencil and paper; sometimes, a computer or calculator. Often a judicious combination is best."
Assessment
Please read the text. You paid too much money to ignore it! Since the reading is so important, some hints on how to do it might be helpful. You may find that slight variations on the following scheme will work for you.
a. Plan on doing the reading more than once, and do not make it an essential goal to understand everything in the reading the first time through it. The first reading should be devoted only to getting a general overview of the material of the section.
b. After the first reading, stop for a few minutes and attempt to summarize to yourself, in your own words, what the section is all about. Then immediately reread the section.
c. During the second reading, make a serious effort to understand all of the material in the section. This does not mean to memorize it, but rather to understand all of the points before going on.
d. If you do not understand something during the second reading, put the book aside awhile and return to it later when your mind is fresher. If you still do not understand it after returning to it, ask me or some other members of the class about it. Do make sure you eventually understand all of the material. You will probably get tripped up in later reading, in doing the homework, or on tests if you treat material you don’t quite understand as "probably not all that important."
e. Do not get discouraged if some points require some time to understand. It is not uncommon to have to think about a point in a math text for a day or even several days before it becomes clear what is really going on.
Throughout the semester you will work on laboratory experiments and projects to help discover and understand some of the basic principles of differential equations. These may be started during a regular class meeting, but you will probably need to complete them outside of class time.
You will receive additional problems sets from the textbook. For those that are collected, you will be given the opportunity to resubmit your solutions as many times as you need until the due date. You are encouraged to work together on these problems, but each student is always expected to write up (and understand) her own solutions.
The problem sets can be found in the Math/Math309 folder on the Student W: drive or at
http://ecademy.agnesscott.edu/Mathematics/mathfacpgs/riddle/courses/math309/
You will have two exams and a final project.
Attendance
Regular attendance for this class will be very important since much of our class time will be spent on discussing problems or working in small groups on problems or computer experiments. It is therefore expected that you will attend and be prepared for every class, but it is also recognized that circumstances may occasionally necessitate missing a class. However, you are still responsible for all material discussed in class whether you are there or not, and for submitting all work before the due date. Approval for extensions must be obtained in advance.
Grading
Your grade will be determined as follows (I reserve the right to make adjustments if necessary).
Your two exams 200 points
Final project 80 points
Problem sets, lab work, and projects 150 points
Class participation 20 points
Total 450 points