A first course in ordinary differential equations together with a begnning look at the simplest second order partial differential equations and their applications.

Table of Contents

Pre-Requisites

Officially, it's only MATH 141, but I truly believe that the prerequisites should be MATH 220 and MATH 230/231. It never fails to amaze me how Penn State can hold its head up and say we actually have a reasonable sequence for these 200 level math courses.

Content

Differential equations are the primary means of connecting mathematics with the world around us. They arise in physics, economics, biology, engineering of all kinds as well as in purely mathematical contexts. Their study is therefore one of the most fruitful parts of the mathematics curriculum.

This course is designed to be an introduction to the basic theory of differential equations. As such its main focus is on the elementary theory of Ordinary Differential Equations - starting with the theory of first order equations and their applications to the life sciences, physics and engineering. Continuing with the theory of linear second order equations with constant coefficients we develop a comprehensive theory of solutions for these together with methods for solving the non-homogeneous problems that occur in the theory of forced vibrations. As a by-product, we also develop the connections of this theory with linear algebra.

Following the theory of second order equations, we move to the study of higher order linear equations with constant coefficients. The connection of this with the method of undetermined coefficents is made as well as the extension of all the methods for second order linear, constant coefficient equations is derived. As not all linear ordinary differential equations of interest have constant coefficients, the next topic is the development of the theory of series soltuions of linear ordinary ordinary differential equations. Series solutions around regular as well as regular singular points are examined.

The last topic in the study of ordinary differential equations is the theory of the Laplace transform. This is one of the most important tools in linear control theory and in signal processing. The course provides an introduction to the elementary part of the theory and its application to the solution of initial value problems of all orders as well as the calculational aspects of the theory.

The last part of the course deals with the elementary parts of the theory of partial differential equations such as the heat, Laplace and wave equations. To reduce the solution of a linear partial differential equation to the theory of ordinary differential equations we introduce the technique of separation of variables. This leads to the theory of Fourier series for building general solutions to initial-boundary value problems out of the basis solutions coming from separation of variables.

Goals

At the end of this course you should be able to:

• Recognize and classify differential equations according to their degree, whether they are linear or nonlinear, whether they are ordinary differential equations or partial differential equations and what properties might be associated with their solutions.
• Find general the solution to linear, separable, homogenous or exact first order ordinary differential equations, as well as solve initial value problems.
• Solve second and higher order, linear, ordinary differential equations with constant coefficients - both homogenous and non-homogeneous using both undetermined coefficients and variation of parameters.
• Analyze oscillatory and forced vibrational phenomena using the methods of the characteristic polynomial and the methods for finding solutions of linear equations.
• Analyze initial value problems for linear differential equations of all orders using the Laplace transform.
• Analyze the solutions of boundary and initial-boundary value problems for the classical partial differential equations of physics suing separation of variables and Fourier series.

Assessment

The grading scheme is the following:

1. Each question you want to get regraded should be on a separate page or pages. Multipart questions can go on the same page.
2. Show all your work for the question.
3. Attach all the regrade pages together using either staples or a paper-clip and slip the regrade into the exam.
4. You must include your original exam. No original, no regrade.

Schedule

Whatever

Syllabi

Summer 2005

Resources

None available, anywhere ever.