Instructor:	David Barsky
Telephone:	750-4201 (with voice mail)
E-mail:		djbarsky@csusm.edu
URL:		http://www.csusm.edu/public/DJBarskyWebs/mainpage.html
Office:		Craven Hall, Room 6231
Office Hours:	To Be Announced

Prerequisite: MATH 162 (Calculus with Applications, II) with a grade of C or better. (Note: a C- is not a grade of C or better.)

I. Catalog Description. Analysis and application of ordinary differential equations: linear and nonlinear equations, existence and uniqueness theorems, analytic methods, qualitative analysis of solutions, numerical methods. Combines theoretical ideas along with hands-on experience using appropriate computer software.

II. Expanded Description. In this course we will cover the following chapters in the course text:

1. Introduction
2. First Order Differential Equations
3. Mathematical Models and Numerical Methods Involving First Order Equations
4. Linear Second Order Equations
5. Applications and Numerical Methods For Second Order Equations and Systems
6. Theory of Higher-Order Linear Differential Equations
8. (if time permits) Seriers Solutions of Differential Equations

There will be several instances where I will not follow the text directly, and I will instead distribute my own lecture notes to you. I will regularly give you reading assignments so that you can tell what material we will be covering next.

In this course we will learn how to solve certain types of problems. A typical problem that we will encounter is the following: suppose that the derivative of a certain function (which we are not allowed to see) satisfies a certain equation (which we are allowed to see, e.g. y'(x) = 2x); can we determine what this hidden function is? In the example given immediately above, we know (from the Fundamental Theorem of Calculus) that there are lots of possible answers, each of which is of the form y(x)=x^2+c. Thus, in a certain sense, you have been solving differential equations ever since you took MATH 160. Indeed, the "techniques of integration" part of MATH 162 was really a second course in differential equations. In this course we will study differential equations which are more complicated than those you saw in MATH 162 because we will not generally be given the derivative as an explicit function of x.

So, you may be wondering, "what's the big deal about differential equations?" The short answer to this question is that the laws of nature are written in the language of differential equations. The long answer is one which you must discover for yourself by taking further courses in math, science and engineering. (Hint: The long answer is the same as the short answer.) Although there may not be enough time to apply everything that we will learn to the real world, it is my hope that you will get at least a glimpse of how differential equations are used to model natural phenomena.

There are many different facets to this course; five primary aspects are described below.

First, this will be a rigorous course in mathematics. That means that we will be interested in much more than just learning how to solve differential equations; we will want to understand solution methods: why do they work the way that they do, when can they go wrong, how can we tell that they are not going to work, and what we should do then?

Second, this will be a modern course in differential equations. Traditional differential equations courses often get a reputation as being "cookbook" courses where you all you do is learn a lot of "recipes" for solving differential equations. By using some ideas from linear algebra (- don't worry if you haven't taken a course in linear algebra; I will not assume any previous familiarity with this material, and I will develop from scratch those parts of the subject that we need) we will see that there really are only a few key ideas underlying most of these solution methods. We will learn how to use both the theoretical side of linear algebra to predict general properties of the solutions to equations, and we will use the more concrete aspects to actually produce solutions.

Third, applications are an integral part of any course on differential equations. Throughout the course we will spend time (both in class and in homework) working on real-world problems. You will be expected to be able to take a real-world problem (i.e., a word problem), formulate it as a mathematics problem, solve that mathematical problem, and decide what the implications that solution has for the original real-world problem. We will find that understanding how to apply differential equations goes hand-in-hand with understanding the theory of differential equations.

Fourth, part of this class is a computer laboratory. We will use a powerful Computer Algebra System (CAS) called Maple running on Macintosh computers to help improve our intuition reagrding the nature of solutions to differential equations. We will also make use of this software when we discuss numerical solution methods. Let me place an important warning here: Maple is not a substitute for thinking! This program can be a valuable tool for gaining insight into a particular problem ... but only if it is used intelligently. It is by learning the theory of differential equations that we can develop the skill to recognize when Maple is giving us answers that we can trust. A useful side-benefit of this laboratory is that developing some familiarity ith Maple should prove advantageous in other Mathematics courses that you take, and possibly later on when you are looking for employment.

Last but definitely not least, like every other course at CSUSM, this class has a 2500 word writing requirement. We will substantially exceed that minimum requirement. In your homework, you will need to do more than just "get the answer in the back of the book." You will need to present your work in a clear, organized fashion so that anyone looking at your solution will be able to understand how you arrived at your conclusion. To repeat the author of the calculus text used at CSUSM:

Part of the aim of this course is to train you to think logically. Learn to write the solutions of the exercises in a connected step-by-step fashion with explanatory words and symbols - not just a string of disconnected equations or formulas.

III. Course Goals. This course is a natural continuation of MATH 162, Calucus with Applications, II. The basic goal of this course is for you to learn how to solve differential equations. To a large extent that means learning various methods for solving different kinds of equations, but actually, the concepts underlying these methods are more important than the techniques themselves, and they will be stressed. Some secondary goals are:

Developing an appreciation of the power of abstraction - seeing how using ideas from linear algebra allows us to develop simple techniques of wide applicability), Learning how mathematics is used to model real-world phenomena, Improving your ability to write mathematics - you should be able (by writing in complete sentences) to explain basic concepts, discuss the meaning of key results, and summarize (parts of) proofs of key theorems, Gaining familiarity with using a Computer Algebra System, and Reinforcing the material that you learned in MATH 162 - we will use integration techniques throughout the course, and in Chapter 7 you will see how useful power series are.

IV. Grades. Your grade will be based on:

	Homework		10%
	Quizzes			18%
	Three Hour Exams	18, 18, and 12%,
	Final Exam		24%

V. Homework. Homework, and its due date, will be posted on the chalkboard (or whiteboard) at lecture. The general rule of thumb is that an assignment made on a Wednesday or Friday will be due the Friday of the following week, while assignments announced on Monday will be due the Friday of that week. I will try to arrange my office hours (a handout listing them will be distributed at the end of the first week of the semester) so that you should always have the opportunity to attempt the homework first, then come see me if you have difficulty, and finally go back and finish the assignment by the due date (if you start it early enough). Late homework will generally not be accepted. If you have to miss a lecture, you should make arrangements to have your homework placed under my door no later than 1:00pm of the day on which it is due; it is your responsibility to get the next homework assignment - you may call me at my office or check the class web site (see Section X). The importance of homework cannot be overstressed: one only learns mathematics by doing it!

As the university writing requirement will be satisfied through the homework, your homework assignments will be graded for both the correctness of your solutions and the level of your exposition. One way to improve the style of your mathematical writing is to study the writing of an accomplished pair of authors: Nagle and Saff. Here are two suggestions that will make you a better writer and thinker of mathematics.
(1) Every time that you sit down to work on the homework, begin by copying one of the examples out of the book. By doing this, you will reap a two-fold benefit. On the one hand, having written it out, you will understand the example better than if you only read it. Additionally, you will become attuned to some of the finer points of mathematical writing, and the style of your writing will improve.
(2) Begin your homework by trying to solve the problems on a piece of scratch paper. After you are satisfied with your conclusion, look over your work and ask yourself, "What did I do to solve this problem?" Your answer (to yourself) should be the starting point for writing up the solution that you will submit for grading. It is important to realize that if you can get the right answer, but you just cannot explain how you did it, then your grasp of the material is extremely shaky and you need to get help quickly (see Section VIII).

I encourage you to form study groups and to work together (this allows you to gain valuable oral communication skills in addition to learning the material) but you must write up your solutions separately (this allows you to gain valuable written communication skills and it keeps you from cheating yourself). Writing up the solutions alone allows you to test yourself to see whether you really understand the material, and this helps to keep you from being "surprised" on the exams. A good strategy is to try first to solve all of the problems yourself, and then meet with some of your classmates who have also already attempted all of the problems to discuss the problems that you have been unable to solve.

I would greatly appreciate it if you would staple your homework pages together, but, please, do not write your solutions in the corner of the paper that will be hidden underneath the staple. Do not use red ink, or a red pencil. I will return the homework after it has been graded before (and after) lecture.

I want to warn you that not all of the homework will be graded; typically only a few representative problems will be examined. This makes it extremely important that you do all of the homework in each set - so that you don't find yourself in the unfortunate situation of having done most of the problems, but not the ones that were graded. Even if this does happen to you once or twice, you should still have an excellent homework grade if you are following the material (if not, get help - see Section IX) and handing in every homework assignment.

VI. Quizzes. Most of our class meetings will begin with a short quiz covering the material of the preceding meeting. To be ready for these quizzes you will need to have reviewed your notes from the preceding lecture (and you must be on-time to class). I will drop your three lowest quiz scores, and any quizzes missed for a valid reason (justified in writing).

VII. Exams. I will be giving three (3) hour exams (well, actually each exam will be almost two hours long) - see Section XI for tentative dates. I will not "drop" any of the exams, but I will count your worst hour exam score as only 12% of your grade; the other scores will each be 18% of your grade. I will not directly assign letter grades to the exams, but I will give some rough indications of how the numerical scores correlate with grades for the sole purpose of helping you evaluate your performance.

If you feel that a problem on your exam was misgraded or unfairly graded, you have one week from the day that exams are handed back to see me about that. After that one week period is over there will be no regrading of problems. (Of course, if the number of points on your exam has been mistallied, that can always be corrected.)

There will be no makeups for the exams. If a single hour exam is missed for a valid reason (justified in writing), then the two remaining exams will each count for 24% of your grade.

The hour exams are tentatively scheduled for Fridays. It is my intent to help you prepare for the exams in the following ways. I will usually distribute a review sheet the Monday before the exam that will help you direct and focus your preparation. At least half of Wednesday's class will be used as a review session, and if necessary, I will schedule an (out-of-class) review period for sometime on Thursday.

Regarding the format of the exams, most exam questions will be similar to homework problems, quiz problems, or examples from lecture and the text, but some others will be less routine and more challenging. There will be problems that require written answers; in your answers to the latter you must use complete sentences and correct punctuation, and your work must be legible. I view the process of preparing for and taking the exams as a "learning experience" for you. Accordingly, I will provide answer sheets for the exams to culminate these experiences.

The best strategy for preparing for these exams is to

	stay current with the course (you need to do this anyway
		in order to maintain your quiz grade),
	do all of the homework (you need to do this anyway
		in order to maintain your homework grade),
	get help (see Section IX) as soon as you realize that you're
		having difficulty with some of the material, and
	get a good night's sleep the night before the exam,
		so that you are rested and ready to think
		when you walk into the classroom.

VIII. Classroom Procedure. I strongly encourage your participation in the lectures. When I ask a question, please don't be afraid to answer it if you think that you can. You probably are right, and even if you aren't, I'm not going to ridicule you. Also, if you have a question (about mathematics) in lecture, please feel free to ask it as it occurs to you. However, I will not answer specific questions about grading in class - see me in my office hours for that. Note also that "Is this going to be on the exam?" is an inappropriate question which will not be answered in the middle of a lecture. When we're working with Maple, if you're having difficulty working out the calculations on the computer, just raise your hand and I'll come over to help you; that's part of why I'm there.

IX. How To Get Help. There are several ways to get help for this course. The main way for you to get help is for you to come see me. I will be running several office hours a week; don't wait for the special review sessions before the exams. You can also contact me by telephone or by e-mail (the number/address is on the first page of this syllabus). Second and third, you can try the Mathematics Learning Assistance Center in Craven 3106-I (phone number 750-4102) and the Academic Support Program for Intellectual Rewards and Enhancement (ASPIRE) in Craven 5201 (phone number 750-4014), but I as this is an upper-division course, I am not certain of how much assistance they can provide.

X. Internet. I will produce and maintain a Web page for this course. The URL is http://www.csusm.edu/public/DJBarskyWebs/362page.html. You can also reach it from the San Marcos home page (http://www.csusm.edu) by going to CSUSM CWIS (the whole thing) [near the top of the page], then to Faculty [under Personal Home Pages, near the bottom of the page], then to David Barsky [about the third screen down], and finally to MATH 362 Differential Equations [about the second screen down]. My MATH 362 page will serve as a repository for official course announcements such as reading assignments, homework assignments, rescheduled hour exams, and (perhaps) capsule summaries of the class lectures. During the one of the lab periods I will show you how to use Netscape Navigator to reach this site, which you can then use as a jumping-off point to reach all sorts of interesting mathematical Web sites (e.g., a site containing over 1000 biographies of mathematicians).

XI. Important Dates. Mark these on your calendar.

First lecture				Monday, January 27
Last day to add classes			Friday, February 7
Last day to drop with no record		Friday, February 7

Last day to change grading option	Friday, February 14
1st Hour Exam (tentatively)		Friday, February 28
Last day to withdraw with a grade of W	Monday, March 17

Spring Break - No class			Monday, March 24
Spring Break - No class			Wednesday, March 26
Spring Break - No class			Friday, March 28

2nd Hour Exam (tentatively)		Friday, April 4
3rd Hour Exam (tentatively)		Friday, May 2
Last class				Friday, May 16

Final Exam (in 211 Academic Hall)	9:45 - 11:45am
					Wednesday, May 21

XIII. Our Contract. By enrolling in and attending this course, you are agreeing to conduct yourself with complete academic honesty at all times. In particular, you are promising to neither receive nor provide assistance on examinations, and the solutions to all of the homework problems that you submit must be composed by yourself. This handout is a contract between you and me. My commitment is to teach the course as described in the preceding sections of this document. By remaining in this course, you affirm that you have read and understood this contract, and that you agree to it.

XIV. Welcome to Differential Equations. - David Barsky

Go back to the main 362 page.