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Ravi P. Agarwal Donal O’Regan

An Introduction to Ordinary Differential Equations

Ravi P. Agarwal Florida Institute of Technology Department of Mathematical Sciences 150 West University Blvd. Melbourne, FL 32901 [email protected]

Donal O’Regan National University of Ireland, Galway Mathematics Department University Road Galway, Ireland [email protected]

ISBN: 978-0-387-71275-8 e-ISBN: 978-0-387-71276-5 DOI: 10.1007/978-0-387-71276-5 Library of Congress Control Number: 2007935982 Mathematics Subject Classification (2000): 34-01, 34-XX ©2008 Springer Science+Business Media LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com

Dedicated to

Ramesh C. Gupta on his 65th birthday

Preface Ordinary differential equations serve as mathematical models for many exciting “real-world” problems, not only in science and technology, but also in such diverse fields as economics, psychology, defense, and demography. Rapid growth in the theory of differential equations and in its applications to almost every branch of knowledge has resulted in a continued interest in its study by students in many disciplines. This has given ordinary differential equations a distinct place in mathematics curricula all over the world and it is now being taught at various levels in almost every institution of higher learning. Hundreds of books on ordinary differential equations are available. However, the majority of these are elementary texts which provide a battery of techniques for finding explicit solutions. The size of some of these books has grown dramatically—to the extent that students are often lost in deciding where to start. This is all due to the addition of repetitive examples and exercises, and colorful pictures. The advanced books are either on specialized topics or are encyclopedic in character. In fact, there are hardly any rigorous and perspicuous introductory texts available which can be used directly in class for students of applied sciences. Thus, in an effort to bring the subject to a wide audience we provide a compact, but thorough, introduction to the subject in An Introduction to Ordinary Differential Equations. This book is intended for readers who have had a course in calculus, and hence it can be used for a senior undergraduate course. It should also be suitable for a beginning graduate course,