• Document: CHAPTER 17 CALCULUS. Based on the original work by. George B. Thomas, Jr. Massachusetts Institute of Technology. as revised by
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CHAPTER 17 THOMAS’ CALCULUS Twelfth Edition Based on the original work by George B. Thomas, Jr. Massachusetts Institute of Technology as revised by Maurice D. Weir Naval Postgraduate School Joel Hass University of California, Davis Copyright © 2010 Pearson Education, Inc. All rights reserved Editor-in-Chief: Deirdre Lynch Senior Acquisitions Editor: William Hoffman Senior Project Editor: Rachel S. Reeve Associate Editor: Caroline Celano Associate Project Editor: Leah Goldberg Senior Managing Editor: Karen Wernholm Senior Production Supervisor: Sheila Spinney Senior Design Supervisor: Andrea Nix Digital Assets Manager: Marianne Groth Media Producer: Lin Mahoney Software Development: Mary Durnwald and Bob Carroll Executive Marketing Manager: Jeff Weidenaar Marketing Assistant: Kendra Bassi Senior Author Support/Technology Specialist: Joe Vetere Senior Prepress Supervisor: Caroline Fell Manufacturing Manager: Evelyn Beaton Production Coordinator: Kathy Diamond Composition: Nesbitt Graphics, Inc. Illustrations: Karen Heyt, IllustraTech Cover Design: Rokusek Design Cover image: Forest Edge, Hokuto, Hokkaido, Japan 2004 © Michael Kenna About the cover: The cover image of a tree line on a snow-swept landscape, by the photographer Michael Kenna, was taken in Hokkaido, Japan. The artist was not thinking of calculus when he composed the image, but rather, of a visual haiku consisting of a few elements that would spark the viewer’s imagination. Similarly, the minimal design of this text allows the central ideas of calculus developed in this book to unfold to ignite the learner’s imagination. For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on page C-1, which is hereby made part of this copyright page. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Addison-Wesley was aware of a trademark claim, the designa- tions have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data Weir, Maurice D. Thomas’ Calculus / Maurice D. Weir, Joel Hass, George B. Thomas.—12th ed. p. cm ISBN 978-0-321-58799-2 1. Calculus—Textbooks. I. Hass, Joel. II. Thomas, George B. (George Brinton), 1914–2006. III. Thomas, George B. (George Brinton), 1914–2006. Calculus. IV. Title V. Title: Calculus. QA303.2.W45 2009b 515–dc22 2009023069 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-848-7047, or e-mail at http://www.pearsoned.com/legal/permissions.htm. 1 2 3 4 5 6 7 8 9 10—CRK—12 11 10 09 ISBN-10: 0-321-58799-5 www.pearsoned.com ISBN-13: 978-0-321-58799-2 Copyright © 2010 Pearson Education, Inc. All rights reserved Chapter SECOND-ORDER 17 DIFFERENTIAL EQUATIONS OVERVIEW In this chapter we extend our study of differential equations to those of second order. Second-order differential equations arise in many applications in the sciences and engineering. For instance, they can be applied to the study of vibrating springs and electric circuits. You will learn how to solve such differential equations by several methods in this chapter. 17.1 Second-Order Linear Equations An equation of the form P(x)y–(x) + Q(x)y¿(x) + R(x)y(x) = G(x), (1) which is line

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