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Principles of Differential and Integral Equations
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AMS Chelsea Publishing
1977; 205 pp; hardcover
Volume: 295
ISBN-10: 0-8218-4622-1
ISBN-13: 978-0-8218-4622-3
List Price: US$32 Member Price: US$28.80
Order Code: CHEL/295.H

In summary, the author has provided an elegant introduction to important topics in the theory of ordinary differential equations and integral equations.

--Mathematical Reviews

This book is intended for a one-semester course in differential and integral equations for advanced undergraduates or beginning graduate students, with a view toward preparing the reader for graduate-level courses on more advanced topics. There is some emphasis on existence, uniqueness, and the qualitative behavior of solutions. Students from applied mathematics, physics, and engineering will find much of value in this book.

The first five chapters cover ordinary differential equations. Chapter 5 contains a good treatment of the stability of ODEs. The next four chapters cover integral equations, including applications to second-order differential equations. Chapter 7 is a concise introduction to the important Fredholm theory of linear integral equations. The final chapter is a well-selected collection of fascinating miscellaneous facts about differential and integral equations.

The prerequisites are a good course in advanced calculus, some preparation in linear algebra, and a reasonable acquaintance with elementary complex analysis. There are exercises throughout the text, with the more advanced of them providing good challenges to the student.

Introduction: Some Elementary Methods
• The linear equation of the first order
• The equation with separable variables
• Exact differential equations
• The uniqueness problem: An example
• Some integral inequalities
• Problems
Existence Theory for Differential Equations
• The first-order equation
• First-order differential systems
• Equations and systems of higher order
• Peano existence theorem
• A uniqueness theorem
• Problems
Some Global Problems for Ordinary Differential Equations
• Statement of the problems
• Global uniqueness
• Global existence and the behavior of saturated solutions
• Dependence of solutions on initial values
• Differential inequalities and the comparison method
• A criterion of global existence
• Problems
Some Special Classes of Differential Systems and Equations
• Linear systems: Generalities
• Linear homogeneous systems
• Nonhomogeneous systems
• Linear equations of higher order
• Autonomous systems
• Linear systems and equations with constant coefficients
• Problems
Stability Theory of Ordinary Differential Systems
• Definitions and examples
• Stability of linear systems
• Stability in the first approximation
• Stability theorems by comparison method
• Further stability results
• Stability of automatic control systems
• Problems
Volterra Integral Equations
• Existence and uniqueness of solution
• An existence theorem
• The linear equation
• The first-kind linear equation
• Some problems on the half-axis
• Problems
Fredholm Theory of Linear Integral Equations
• The resolvent kernel
• The entire functions of Fredholm and their applications
• A glimpse of Hilbert space theory
• Eigenvalues, eigenfunctions and adjoint equations
• Problems
Theory of Self-Adjoint Integral Equations and Some Applications
• Some properties of an integral operator
• The existence of eigenvalues
• The Hilbert-Schmidt expansion theorem
• Complete kernels and systems
• The Sturm-Liouville problem
• Reduction to an integral equation: The first case
• Reduction to an integral equation: The second case
• Problems
Miscellanea (Nice Things)
• Maximum and minimum solutions of scalar differential equations
• A theorem of Massera
• An oscillation theorem
• A formula of Picone and its consequences
• Liénard's equation
• Differential systems without uniqueness
• Abel's integral equation
• References
• Index