Department of Mathematics

Syllabus

**Math 01.231 - Ordinary Differential Equations**

**CATALOG DESCRIPTION:**

Math 01.231 Ordinary Differential Equations
3 s.h.

(Prerequisites: Math 01.230 Calculus III and Math 01.210 Linear Algebra with a C- or better in both courses)

Applications of ordinary differential equations and their methods of
solution form the major part of this course. It also includes the
solution of nth order equations, particularly of first and higher degree
linear differential equations, and series and Laplace Transform solutions.
Students can be asked to use computers and/or graphing calculators as an
aid in solving equations.

**OBJECTIVES:**

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.

**CONTENT:**

1.0 **Equations of Order One**

1.1 The isoclines of an equation

1.2 An existence theorem

1.3 Separation of variables

1.4 Homogeneous functions

1.5 Equations with homogeneous coefficients

1.6 Exact equations

1.7 The linear equation of order one

2.0 **Elementary Applications**

2.1 Velocity of escape from the earth

2.2 Orthogonal trajectories

3.0 **Linear Differential Equations**

3.1 The general linear equation

3.2 Linear independence

3.3 An existence and uniqueness theorem

3.4 The Wronskian

3.5 General solution of a homogeneous linear equation

3.6 General solution of a nonhomogeneous linear equation

3.7 Differential operators

4.0 **Linear Equations with Constant Coefficients**

4.1 Introduction

4.2 The auxiliary equation; distinct roots

4.3 The auxiliary equation; repeated roots

4.4 A definition of exp z for imaginary z

4.5 The auxiliary equation; imaginary roots

5.0 **Nonhomogeneous Equations: Undetermined Coefficients**

5.1 Construction of a homogeneous equation from a specified solution

5.2 Solution of a nonhomogeneous equation

5.3 The method of undetermined coefficients

5.4 Solution by inspection

6.0 **The Laplace Transform**

6.1 The transform concept

6.2 The definition of the Laplace transform

6.3 Transforms of elementary functions

6.4 Sectionally continuous functions

6.5 Functions of exponential order

6.6 Functions of class A

6.7 Transforms of derivatives

6.8 Derivatives of transforms

6.9 The gamma function

6.10 Periodic functions

7.0 **Inverse Transforms**

7.1 Definition of an inverse function

7.2 Partial fractions

7.3 Initial value problems

8.0 **Power Series Solutions**

8.1 Linear equations and power series

8.2 Convergence of power series

8.3 Ordinary points and singular points

8.4 Validity of the solutions near an ordinary point

8.5 Solutions near an ordinary point

9.0 **System of Differential Equations**

9.1 Systems and Techniques

9.2 Applications

10.0 **Numerical Techniques**

10.1 Euler-Cauchy Method

10.2 Adams-Bashforth Method

10.3 Runze-Kutta Method

10.4 Milne's Method

10.5 The Method of Successive Approximations

**TEXTS:**

*Boyce, William E., DiPrima, Richard ELEMENTARY DIFFERENTIAL EQUATIONS AND BOUNDARY PROBLEMS, 8th ed., Wiley & Sons Inc., New York.

Zill, Dennis A FIRST COURSE IN DIFFERENTIAL EQUATIONS with modeling applications, 6th ed., Brooks/Cole, London, 1997.

Borrelli, Robert L., and Coleman, Courtney S. DIFFERENTIAL EQUATIONS A Modeling Perspective, Wiley & Sons Inc., New York, 1998.

Blanchard, Paul, Devaney, Robert L., and Hall, Glen R. DIFFERENTIAL EQUATIONS, Brooks/Cole, Pacific Grove, 1998.