College of the Holy Cross Mathematics and Computer Science
College of the Holy Cross Mathematics and Computer Science
Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred), or jlittle@holycross.edu
Office Hours: TBA
Course Homepage: http://math.holycross.edu/~little/ODE00/ODE.html
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)
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 (If that is not feasible, we might try at least 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 the study of certain simple electric 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
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 existence and uniqueness theorems for solutions of ordinary differential equations; graphical, analytic, and numerical solution techniques for first- and second-order equations with one unknown function; matrix methods for linear first order systems and higher-order equations.
The rough schedule for the semester is as follows:
This course forms the first half of a new two-semester sequence in Differential Equations, MATH 304, 373. This combination will satisfy the full-year linked-sequence course requirement for the Mathematics major. The second semester (covering partial differential equations, boundary value problems, Fourier series techniques, and Sturm-Liouville theory) will be substantially the same as large portions of the previous Applied Mathematics 1 course and a portion Applied Mathematics 2. So if you have already taken MATH 373 you will not be able to continue to the second semester this year. Also, only this section (not Professor Anderson's section) will be continuing.
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.
The text for the course is Fundamentals of Differential Equations and Boundary Value Problems, 3rd edition, by R. Kent Nagle, Edward B. Saff, and Arthur D. Snider. We will cover material from chapters 1,2,3,4,5,9 this semester.
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 present and 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. \bigskip Some of the other meetings of the class will be structured as lectures or computer laboratory days when that seems appropriate.
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.
The assignments for the course will consist of:
If you ever have a question about the grading policy, or about your standing in the course, please feel free to consult with me.