College of the Holy Cross Mathematics and Computer Science


Mathematics 304 -- Ordinary Differential Equations

Syllabus, Fall 2004

Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred), or jlittle@holycross.edu
Office Hours: MW 1-3pm, TR 9-11am, F 10-11am, and by appointment
Course Homepage: http://math.holycross.edu/~little/ODE04/ODEHome.html

Course Description

A differential equation is a relation between derivatives of an unknown function (often including the "zero-th" derivative -- the unknown function itself) and other known functions. If the unknown function depends on just one variable, the differential equation is said to be an ordinary differential equation (or ODE for short). Differential equations involving unknown functions of more than one variable and partial derivatives are called partial differential equations. The order of a differential equation is the highest order derivative of the unknown function that is present. For example, if y = y(t) is a function of one variable and m, k, b, F, w are all constant, then

(1)

m d2y/dt2 + k dy/dt + by = F cos(wt)

is an example of an ordinary differential equation of second order. We will study equations of this type in great detail in this course.

In almost all cases, we will be interested in solving differential equations to determine which functions satisfy the relation, or if that is not feasible, we might at least try to use the equation to derive some qualitative information about solutions.

Differential equations are sometimes studied "in the abstract" in mathematics. But the true importance of this subject comes from the fact that many of the most important and successful techniques for modeling physical and biological phenomena are based on differential equations. Indeed, it is no exaggeration to say that understanding of differential equations, developed starting with the work of Newton and Leibniz on the foundations of the calculus and continuing to the present, has formed the basis for a large portion of modern science and technology. The underlying reason for this is that many physical "laws" and patterns that scientists have observed take the form of a relation between rates of change (that is, derivatives) of quantities and the quantities themselves. Thus we obtain differential equations if the relation is stated in mathematical terms. For example, differential equations of the form (1) above arise in the study of damped, forced harmonic oscillators and also in the study of some simple electrical circuits.

For another example, in decay of radioactive isotopes, the rate of change of the amount of the radioactive substance is proportional to the amount at all times. In mathematical terms, if y(t) represents the amount at time t, then we obtain the relation

dy/dt = ky,

the familiar first order exponential decay/growth equation. The solutions of this equation are the functions y(t) = y(0)ekt. Knowing this lets us predict the amount present at future times provided we know the initial amount, y(0), and the decay rate constant, k. In a similar way, if we know the underlying relationship governing a physical process and we can solve the corresponding differential equation, then we can predict what will happen as time goes on and even try to control the process in some cases. (Control becomes a concern if it is possible to adjust parameters like the constant k in the equation above to try to affect the behavior of solutions. This would not be possible in radioactive decay of a specific isotope, of course. But it is possible in other situations described by other equations.)

In this course, we will study graphical, analytic, and numerical solution techniques for first-order equations with one unknown function; matrix methods for linear first order systems and higher-order equations, the ``dynamical'' classification of the behavior of solutions of these systems, existence and uniqueness theorems for solutions of ordinary differential equations, and qualitative results on behavior of solutions of nonlinear systems.

The rough schedule for the semester is as follows:

The remaining 2 days will be devoted to in-class midterm exams. A more detailed day-by-day schedule is available on the course homepage.

This course forms the first half of a two-semester sequence in Differential Equations; the second half is MATH 373, Applied Mathematics. This combination will satisfy the full-year linked-sequence course requirement for the Mathematics major, and both courses fall in the Applied Mathematics distribution area for the major. The second semester covers partial differential equations, boundary value problems, Fourier series techniques, and Sturm-Liouville theory. Important Note: If you are expecting to take both semesters of the sequence, you should know that only Professor Roberts' section will be continuing in the Spring.

A Comment About This Course

In addition to drawing on your conceptual understanding of the mathematics you have seen so far in your college work, I expect that you will find that this course makes very heavy use of several specific computational topics from calculus (methods of integration such as substitution, integration by parts, some partial fractions) and linear algebra (computing eigenvalues and eigenvectors of matrices). You will need to be able to carry out the relevant processes symbolically by hand to complete many of the problems that will be assigned. You may need to "brush up" on these topics when we get into the sections of the course where these methods are used. You will probably want to refer to your college calculus and linear algebra texts as references. If you no longer have copies of those books, you can refer to the texts on reserve for this course in the Science Library as needed.

Text

The text for the course is Differential Equations, Dynamical Systems, & and Introduction to Chaos, 2nd edition, by Morris W. Hirsch, Stephen Smale, and Robert L. Devaney. We will cover material from chapters 1-12 this semester. At a couple of points, Xeroxed notes will be distributed for topics not covered in detail in the text.

Course Format

To get you more directly involved in the subject matter of this course, several times during the semester, the class will break down into groups of 3 or 4 students for one or more days, and each group will work together for a portion of those class periods on a group discussion exercise. The exercises will be made up by me. I will be available for questions and other help during these periods. At the conclusion of some of these discussions, at times the class as a whole may reconvene to talk about what has been done, to sum up the results, to hear short oral reports from each group, etc. Each group will be responsible for a write-up of solutions for the questions from each discussion day, and those will be graded and and returned with comments. The other meetings of the class will be structured as lectures or computer laboratory days when that seems appropriate.

Computer Work

We will be using Maple on the departmental Sun workstation network quite extensively throughout the course to implement the techniques we discuss and to generate solutions to problems. Several class meetings will take place in the SW 219 computer lab and some of the individual problem sets will include problems for which you will need to use Maple.

Grading

The assignments for the course will consist of:

  1. Two in-class midterm exams, each worth 15% of course grade. The (tentative) dates are Friday, October 1 and Friday, November 19.
  2. Final exam worth 25% of the course grade. The final exam for this course will be given Thursday, December 16, at 8:30 a.m..
  3. Weekly individual problem sets, worth 25% of the course grade.
  4. Group reports from discussion and computer lab days, worth 20% of the course grade.

If you ever have a question about the grading policy, or about your standing in the course, please feel free to consult with me.

Departmental Statement on Academic Integrity


Why is academic integrity important?


All education is a cooperative enterprise between teachers and students. This cooperation works well only when there is trust and mutual respect between everyone involved. One of our main aims as a department is to help students become knowledgeable and sophisticated learners, able to think and work both independently and in concert with their peers. Representing another person's work as your own in any form (plagiarism or ``cheating''), and providing or receiving unauthorized assistance on assignments (collusion) are lapses of academic integrity because they subvert the learning process and show a fundamental lack of respect for the educational enterprise.

How does this apply to our courses?


You will encounter a variety of types of assignments and examination formats in mathematics and computer science courses. For instance, many problem sets in mathematics classes and laboratory assignments in computer science courses are individual assignments. While some faculty members may allow or even encourage discussion among students during work on problem sets, it is the expectation that the solutions submitted by each student will be that student's own work, written up in that student's own words. When consultation with other students or sources other than the textbook occurs, students should identify their co-workers, and/or cite their sources as they would for other writing assignments. Some courses also make use of collaborative assignments; part of the evaluation in that case may be a rating of each individual's contribution to the group effort. Some advanced classes may use take-home examinations, in which case the ground rules will usually allow no collaboration or consultation. In many computer science classes, programming projects are strictly individual assignments; the ground rules do not allow any collaboration or consultation here either.

What are the responsibilities of faculty?


It is the responsibility of faculty in the department to lay out the guidelines to be followed for specific assignments in their classes as clearly and fully as possible, and to offer clarification and advice concerning those guidelines as needed as students work on those assignments. The Department of Mathematics and Computer Science upholds the College's policy on academic honesty. We advise all students taking mathematics or computer science courses to read the statement in the current College catalog carefully and to familiarize themselves with the procedures which may be applied when infractions are determined to have occurred.

What are the responsibilities of students?


A student's main responsibility is to follow the guidelines laid down by the instructor of the course. If there is some point about the expectations for an assignment that is not clear, the student is responsible for seeking clarification. If such clarification is not immediately available, students should err on the side of caution and follow the strictest possible interpretation of the guidelines they have been given. It is also a student's responsibility to protect his/her own work to prevent unauthorized use of exam papers, problem solutions, computer accounts and files, scratch paper, and any other materials used in carrying out an assignment. We expect students to have the integrity to say ``no'' to requests for assistance from other students when offering that assistance would violate the guidelines for an assignment.

Specific Guidelines for this Course


In this course, all examinations and quizzes will be closed-book and given in-class. No sharing of information with other students in any form will be permitted during exams and quizzes. On group discussion write-ups, close collaboration is expected. On the problem sets, discussion of the questions with other students in the class, and with me during office hours is allowed, even encouraged. BUT, if you do take advantage of any of these options, you will be required to state that fact in a "footnote" accompanying the problem solution. Failure to follow this rule will be treated as a violation of the College's Academic Integrity policy.