An Introduction to Ordinary Differential Equations

by Earl A. Coddington

Overview

"Written in an admirably cleancut and economical style." — Mathematical Reviews.
This concise text offers undergraduates in mathematics and science a thorough and systematic first course in elementary differential equations. Presuming a knowledge of basic calculus, the book first reviews the mathematical essentials required to master the materials to be presented.
The next four chapters take up linear equations, those of the first order and those with constant coefficients, variable coefficients, and regular singular points. The last two chapters address the existence and uniqueness of solutions to both first order equations and to systems and n-th order equations.
Throughout the book, the author carries the theory far enough to include the statements and proofs of the simpler existence and uniqueness theorems. Dr. Coddington, who has taught at MIT, Princeton, and UCLA, has included many exercises designed to develop the student's technique in solving equations. He has also included problems (with answers) selected to sharpen understanding of the mathematical structure of the subject, and to introduce a variety of relevant topics not covered in the text, e.g. stability, equations with periodic coefficients, and boundary value problems.

Product Details

ISBN-13:	9780486659428
Publisher:	Dover Publications
Publication date:	03/01/1989
Series:	Dover Books on Mathematics
Edition description:	Unabridged
Pages:	320
Product dimensions:	5.50(w) x 8.50(h) x (d)

Table of Contents

Chapter 0. Preliminaries
1. Introduction
2 Complex numbers
3 Functions
4 Polynomials
5. Complex series and the exponential function
6. Determinants
7. Remarks on methods of discovery and proof
Chapter 1. Introduction—Linear Equations of the First Order
1. Introduction
2. Differential equations
3. Problems associated with differential equations
4. Linear equations of the first order
5. The equation y'+ay=0
6. The equation y'+ay=b(x)
7. The general linear equation of the first order
Chapter 2. Linear Equations with Constant Coefficients
1. Introduction
2. The second order homogeneous equation
3. Initial value problems for second order equations
4. Linear dependence and independence
5. A formula for the Wronskian
6. The non-homogeneous equation of order two
7. The homogeneous equation of order n
8. Initial value problems for n-th order equations
9. Equations with real constants
10. The non-homogeneous equation of order n
11. A special method for solving the non-homogeneous equation
12. Algebra of constant coefficient operators
Chapter 3. Linear Equations with Variable Coefficients
1. Introduction
2. Initial value problems for the homogeneous equation
3. Solutions of the homogeneous equation
4. The Wronskian and linear independence
5. Reduction of the order of a homogeneous equation
6. The non-homogeneous equation
7. Homogeneous equations with analytic coefficients
8. The Legendre equation
9. Justification of the power series method
Chapter 4. Linear Equations with Regular Singular Points
1. Introduction
2. The Euler equation
3. Second order equations with regular singular points—an example
4. Second order equations with regular singular points—the general case
5. A convergence proof
6. The exceptional cases
7. The Bessel equation
8. The Bessel equation (continued)
9. Regular singular points at infinity
Chapter 5. Existence and Uniqueness of Solutions to First Order Equations
1. Introduction
2. Equations with variables separated
3. Exact equations
4. The method of successive approximations
5. The Lipschitz condition
6. Convergence of the successive approximations
7. Non-local existence of solutions
8. Approximations to, and uniqueness of, solutions
9. Equations with complex-valued functions
Chapter 6. Existence and Uniqueness of Solutions to Systems and n-th Order Equations
1. Introduction
2. An example—central forces and planetary motion
3. Some special equations
4. Complex n-dimensional space
5. Systems as vector equations
6. Existence and uniqueness of solutions to systems
7. Existence and uniqueness for linear systems
8. Equations of order n
References; Answers to Exercises; Index