Differential Equations and Their Applications An Introduction to App
This textbook is a unique blend of the theory of differential equations and their exciting application to "real world" problems. First, and foremost, it is a rigorous study of ordinary differential equations and can be fully un derstood by anyone who has
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M. Braun
Differential Equations and Their Applications An Introduction to Applied Mathematics 2nd Edition
Springer-Verlag
New York· Heidelberg· Berlin
Martin Braun Department of Mathematics Queens College City University of New Y ork Flushing, NY 11367 USA
Editors
Fritz John
Lawrence Sirovich
Courant Institute of Mathematical Studies New York University New York, NY 10012 USA
Division of Applied Mathematics Brown University Providence, RI 02912 USA
Joseph P. LaSalle
Gerald B. Whitham
Division of Applied Mathematics Brown University Providence, RI 02912 USA
Applied Mathematics Firestone Laboratory California Institute of Technology Pasadena, CA 91125 USA
AMS Subject Classifications: 98A20, 98A35, 34-01
Library of Congress Cataloging in Publication Data Braun, Martin, 1941Differential equations and their applications. (Applied mathematical sciences ; v. 15) Includes index. 1. Differential equations. I. Title. 11. Series. 5\O'.8s [515'.35] QAI.A647 vol. 15 1978 [QA37I] 77-8707
All righ ts reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1975, 1978 by Springer-Verlag, New York Inc. Softcover reprint ofthe hardcover 2nd edition 1978
9 8 7 6 543 2 1 ISBN 978-0-387-90266-1 ISBN 978-1-4684-9360-3 (eBook) D0110.1007/978-1-4684-9360-3
To four beautiful people:
Zelda Lee Adeena Rachelle, I. Nasanayl, and Shulamit
Preface
This textbook is a unique blend of the theory of differential equations and their exciting application to "real world" problems. First, and foremost, it is a rigorous study of ordinary differential equations and can be fully understood by anyone who has completed one year of calculus. However, in addition to the traditional applications, it also contains many exciting "real life" problems. These applications are completely self contained. First, the problem to be solved is outlined clearly, and one or more differential equations are derived as a model for this problem. These equations are then solved, and the results are compared with real world data. The following applications are covered in this text. I. In Section 1.3 we prove that the beautiful painting "Disciples of Emmaus" which was bought by the Rembrandt Society of Belgium for $170,000 was a modem forgery. 2. In Section 1.5 we derive differential equations which govern the population growth of various species, and compare the results predicted by our models with the known values of the populations. 3. In Section 1.6 we derive differential equations which govern the rate at which farmers adopt new innovations. Surprisingly, these same differential equations govern the rate at which technological innovations are adopted in such diverse industries as coal, iron and steel, brewing, and railroads. 4. In Section 1.7 we try to determine whether tightly sealed drums filled with concentrated waste material will crack upon impact with the ocean floor. In this section we also describe several tricks for obtaining information about solutions of a differential equ
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