06110290: Ordinary Differential Equation (Spring 2008)

Department of Mathematics

Zhejiang University

Announcements    General     Course Goals   Course Description     Topics     Texts     Grading      Syllabus      Assignments  

Announcements

• Feb. 21: The website is open. Look for your ID number.

General
　

Course Goals

In few areas of college mathematics is the interaction of science and mathematics so marked as in the study of differential equations. The purpose of this course is to introduce the student not only to the theoretical aspects of differential equations, including the establishment of existence of solutions, but also to techniques for obtaining solutions for the various types of ordinary differential equations.

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 (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.

Topics
　

This course provides a comprehensive introduction to ordinary differential equations. Topics include:
    ·Classical methods of solving first order and linear higher order ordinary differential equations.
    ·Laplace Transform and Power Series solutions of linear ordinary differential equations.
    ·Matrix solutions to linear systems of ordinary differential equations.
    ·Numerical Methods of solution of first and higher order differential equations.
　

Texts
　

Required textbook:
            《常微分方程》，蔡遂林编著，浙江大学出版社.
　

Comments: 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.

Grading

Credit toward the semester grade will be allocated to each of the components as indicated in the following table.

Note: Final examination will be in-class, closed-book. More information will be provided prior to it. If you ever have a question about the grading policy, or about your standing in the course, please feel free to consult with me.

Note: Here you can view or download the notes that we use in class. DO NOT depend solely on these notes as many details are missing. You should read the textbook and take notes in class.

• See previous course: Spring, 2005-2006

Homework

• Hand in homework before class in every week.
• Please write down your unique ID number on the upper left corner of your homework book.