Fundamentals of differential equations and boundary value problems. PDF
Preview Fundamentals of differential equations and boundary value problems.
SEVENTH EDITION FU N DAM E N TALS O F Get the Most Out of Differential MyMathLab® Equations MyMathLab is the leading online homework, tutorial, and assessment aDF program designed to help you learn and understand mathematics. n U d i fN B ofD and Boundary Value Problems e S Personalized and adaptive learning u A n rM d S e Interactive practice with immediate feedback a E r ynN S Multimedia learning resources V N AG LE | S A FF | S N I D E R tT a iA S Complete eText lua L e lS S Mobile-friendly design P rEO o bqF l eu m a MyMathLab is available for this textbook. s t To learn more, visit www.mymathlab.com i o n s Pearson is the proud sponsor of the International Conference on Technology in Collegiate Mathematics. Please visit www.ictcm.com SEVENTH www.pearsonhighered.com EDITION ISBN-13: 978-0-321-97710-6 ISBN-10: 0-321-97710-6 9 0 0 0 0 NAGLE SAFF SNIDER 9 780321 977106 This page intentionally left blank A01_MISH4182_11_GE_FM.indd 6 10/06/15 11:46 am Get the Most Out of ® MyMathLab : ® MyMathLab Support You Need, When You Need It MyMathLab is the world’s leading online program in mathematics that has helped millions of students succeed in their math courses. Take advantage of the resources it provides. Just-in-time help MyMathLab’s interactive exercises mirror those in the textbook but are programmed to allow you unlimited practice, leading to mastery. Most exercises include learning aids such as “Help Me Solve This,” “View an Example,” and “Tutorial Video,” and (cid:87)(cid:75)(cid:72)(cid:92)(cid:3)(cid:82)(cid:909)(cid:72)(cid:85)(cid:3)(cid:75)(cid:72)(cid:79)(cid:83)(cid:73)(cid:88)(cid:79)(cid:3)(cid:73)(cid:72)(cid:72)(cid:71)(cid:69)(cid:68)(cid:70)(cid:78)(cid:3)(cid:90)(cid:75)(cid:72)(cid:81)(cid:3)(cid:92)(cid:82)(cid:88)(cid:3) enter incorrect answers. Tutorial Video support Instructional videos narrated by the author cover key examples from the text and can conveniently be played on any mobile device. These videos are especially helpful if you miss a class or just need another explanation. Interactive eText The Pearson eText gives you access to your textbook anytime, anywhere. In addition to letting you take notes, highlight, and book- (cid:80)(cid:68)(cid:85)(cid:78)(cid:15)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:51)(cid:72)(cid:68)(cid:85)(cid:86)(cid:82)(cid:81)(cid:3)(cid:72)(cid:55)(cid:72)(cid:91)(cid:87)(cid:3)(cid:82)(cid:909)(cid:72)(cid:85)(cid:86)(cid:3)(cid:76)(cid:81)(cid:87)(cid:72)(cid:85)(cid:68)(cid:70)(cid:87)(cid:76)(cid:89)(cid:72)(cid:3) links throughout, so you can watch videos as you read. www.mymathlab.com A00_NAGL7106_07_SE_EP.indd 3 31/10/16 1:21 PM This page intentionally left blank A01_MISH4182_11_GE_FM.indd 6 10/06/15 11:46 am A BRIEF TABLE OF INTEGRALS* (continued) 1u sin u du = sin u - u cos u. 1un sin u du = -un cos u + n1un-1 cos u du. 1u cos u du = cos u + u sin u. 1un cos u du = un sin u - n1un-1 sin u du. eau a sin nu - n cos nu eau a cos nu + n sin nu 1eau sin nu du = . 1eau cos nu du = . a2 + n2 a2 + n2 1 2 1 2 sin a + b u sin a - b u 1sin au sin bu du = - + , a2 ≠ b2. 2 a + b 2 a - b 1 2 1 2 sin 1a + b 2u sin 1a - b 2u 1cos au cos bu du = + , a2 ≠ b2. 2 a + b 2 a - b 1 2 1 2 c1os a +2b u c1os a -2b u 1sin au cos bu du = - - , a2 ≠ b2. 2 a + b 2 a - b 1 2 1 2 1sinh u du = cosh u. 1 2 1 2 1cosh u du = sinh u. ∞ Γ t = e-uut-1du, t 7 0; Γ 1 = p; and Γ n + 1 = n!, if n is a positive integer. 2 L 0 2 1 2 1 2 1 2 SOME POWER SERIES EXPANSIONS f ″ a f n a f x = f a + f′ a x - a + 2! x - a 2 + g + 1 n2! x - a n + g Taylor series 1 2 1 2 1 2 1 2 1 21 2 1 2 1 2 1 2 ∞ xn ∞ -1 nx2n+1 ∞ -1 nx2n ex = a sin x = a cos x = a n=0 n! n=01 2n2+ 1 ! n=01 2n2 ! ∞ 1 ∞ 2 1 ∞2xn 1 - x -1 = axn 1 - x -2 = a n + 1 xn ln 1 - x = - a n n=0 n=0 n=1 1 2 1 2 1 2 1 2 tan x = x + 13 x3 + 125 x5 + 31175 x7 + 268235 x9 + g ∞ x2n+1 arcsin x = x + x#3 + 1# #3# x5 + 1# #3# #5# x7 + g arctan x = na=0 -1 n2n + 1 2 3 2 4 5 2 4 6 7 1 2 ∞ -1 kx2k ∞ -1 kx2k+1 ∞ -1 kx2k+n J0 x = ka=0 1k! 2222k J1 x = ka=0 k!1k +21 !22k+1 Jn x = ka=0 k!Γ 1n + k2+ 1 22k+n 1 2 1 2 1 2 1 2 1 2 1 2 *Note: An arbitrary constant is to be added to each indefinite integral. A00_NAGL7106_07_SE_EP.indd 5 31/10/16 1:21 PM LINEAR FIRST-ORDER EQUATIONS The solution to y′ + P x y = Q x taking the value y x0 at x = x0 is x y x = e-Lxx0 P j1d2j eL1xj0 2P z dz Q j dj + y 1x0 2 . L 1 2 x0 1 2 1 2 c 1 2 1 2d METHOD OF UNDETERMINED COEFFICIENTS To find a particular solution to the constant-coefficient differential equation ay″ + by′ + cy = Pm t ert , where P t is a polynomial of degree m, use the form m 1 2 yp 1t 2= ts Amtm + g + A1t + A0 ert ; if r is no1t a2 root o1f the associated auxiliary2 equation, take s = 0; if r is a simple root of the associated auxiliary equation, take s = 1; and if r is a double root of the associated auxiliary equation, take s = 2. To find a particular solution to the differential equation ay″ + by′ + cy = Pm t eat cos bt + Qn t eat sin bt , b ≠ 0 , where P t is a polynomial of degree m and Q t is a polynomial of degree n, use the m 1 2 1 2n form 1 2 1 2 yp t = ts Aktk + g + A1t + A0 eat cos bt 1 2 +1ts Bktk + g + B1t +2B0 eat sin bt , where k is the larger1 of m and n. If a + ib is2 not a root of the associated auxiliary equa- tion, take s = 0; if a + ib is a root of the associated auxiliary equation, take s = 1. VARIATION OF PARAMETERS FORMULA If y1 and y2 are two linearly independent solutions to ay″ + by′ + cy = 0, then a particular solution to ay″ + by′ + cy = f is y = y1y1 + y2y2, where -f t y2 t f t y1 t y1 t = LaW 1y12, y21 2t dt, y2 t = LaW1y21, y12 2t dt, 1 2 1 2 and W y1, y2 t = y1 t3y2= t 4-1 2y1= t y2 t . 3 41 2 3 41 2 1 2 1 2 1 2 1 2 A00_NAGL7106_07_SE_EP.indd 6 31/10/16 1:21 PM A TABLE OF LAPLACE TRANSFORMS f t F s = ℒ f s f t F s = ℒ f s 1 2 1 2 5 6 1 2 1 2 1 2 5 6 1 2 1 s 1 p 1. f at F 20. a a t 2s 1 2 a b 2 2 p 2. eat f t F s-a 21. t 22s3/2 2 1 2 1 2 # # 3. f′ t sF s -f 0 22. tn- 1 2 , n = 1, 2,c 1 3 5g 2n-1 p 2nsn+ 1/2 2 1 > 2 1 2 1 2 1 2 1 2 Γ r+1 1 2 4. f n t snF s -sn-1f 0 -sn-2f′ 0 23. tr, r7 -1 sr+1 1 2 1 2 1 2 1 2 1 2 1 2 b - g- sf n-2 0 -f n-1 0 24. sin bt s2+b2 1 2 1 2 5. tn f t -1 nF n s 1 2 1 2 25. cos bt s s2+b2 1 2 1 1 2 1 2 1 2 b 6. t f t 1s∞F u du 26. eat sin bt s-a 2+b2 7. 10t 1 f 2y dy Fss 1 2 27. eat cos bt 1s-sa-22a+b2 1 2 1 2 b 8. f * g t F s G s 28. sinh bt 1 2 s2-b2 1 21 2 110Te2-stf1t2dt s 9. f t+T = f t 29. cosh bt 1-e1-sT2 s2-b2 1 2 1 2 2b3 10. f t-a u t-a , aÚ0 e-asF s 30. sin bt-bt cos bt s2+b2 2 1 2 1 2 1 2 2bs 11. g t u t-a , aÚ0 e-as ℒ g t+a s 31. t sin bt 1 2 s2+b2 2 1 2 1 2 e-as 5 1 261 2 2bs2 12. u t-a , aÚ0 32. sin bt+bt cos bt 1 2 s s2+b2 2 13. q1 a,b t2, 06a6b e-sa-s e-sb 33. t cos bt 1ss22+-bb2222 1 2 4b3 14. d t-a , aÚ0 e-as 34. sin bt cosh bt- cos bt sinh bt 1 2 s4+4b4 1 2 1 2b2s 15. eat 35. sin bt sinh bt s-a s4+4b4 n! 2b3 16. tn, n = 1, 2,c 36. sinh bt- sin bt sn+1 s4-b4 n! 2b2s 17. eattn, n = 1, 2,c 37. cosh bt- cos bt s-a n+1 s4-b4 18. eat-ebt 1s-1aa2-sb2-b 38. Jv bt , v7 -1 12bvs2+s2b+2-b2s2v a-b s 1 2 2 19. aeat-bebt 1 21 2 s-a s-b 1 2 1 21 2 A00_NAGL7106_07_SE_EP.indd 7 31/10/16 1:21 PM SEVENTH EDITION Fundamentals of Differential Equations and Boundary Value Problems R. Kent Nagle Edward B. Saff Vanderbilt University Arthur David Snider University of South Florida A01_NAGL7106_07_SE_FM_i-xx.indd 1 31/10/16 1:21 PM Director, Portfolio Management: Deirdre Lynch Cover is detail of Martin Burgess’ Executive Editor: Jeff Weidenaar “Clock B” that was constructed to Editorial Assistant: Jennifer Snyder demonstrate the efficacy of John Content Producer: Patty Bergin Harrison’s (1693-1776) science. Managing Producer: Karen Wernholm Begun in 1974, and completed by Media Producer: Erin Carreiro the Charles Frodsham & Co. Ltd. MathXL Content Manager: Kristina Evans of London in 2011, Clock B was Product Marketing Manager: Yvonne Vannatta placed on time trial at the Royal Field Marketing Manager: Evan St. Cyr Observatory, Greenwich, England Marketing Assistant: Jennifer Myers where it remained within 5/8 Senior Author Support/Technology Specialist: Joe Vetere seconds in 100 days, and officially Rights and Permissions Project Manager: Gina Cheselka dubbed “the world’s most accurate Manufacturing Buyer: Carol Melville, LSC Communications pendulum clock operating in free Associate Director of Design: Blair Brown air.” The clock exhibits oscillation Composition: Cenveo consistent with van der Pol’s Text Design, Production Coordination, nonlinear differential equation Composition, and Illustrations: Cenveo with an amplitude that minimizes Cover Design: Cenveo external perturbation effects. Cover Image: Donald J. Saff http://burgessclockb.com/ Photo courtesy of: Charles Frodsham & Co. Ltd. Cover photo courtesy of: Donald Saff Copyright © 2018, 2012, 2008 by Pearson Education, Inc. All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information regarding permissions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit www.pearsoned.com/permissions/. PEARSON, ALWAYS LEARNING, and MYMATHLAB are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the U.S. and/or other countries. Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc. or its affiliates, authors, licensees or distributors. Library of Congress Cataloging-in-Publication Data Names: Nagle, R. Kent. | Saff, E. B., 1944- | Snider, Arthur David, 1940- Title: Fundamentals of differential equations and boundary value problems. Description: Seventh edition / R. Kent Nagle, Edward B. Saff, Vanderbilt University, Arthur David Snider, University of South Florida. | Boston : Pearson, [2018] | Includes index. Identifiers: LCCN 2016030692| ISBN 9780321977106 (hardcover) | ISBN 0321977106 (hardcover) Subjects: LCSH: Differential equations—Textbooks. | Boundary value problems—Textbooks. Classification: LCC QA371 .N243 2018 | DDC 515/.35—dc23 LC record available at https://lccn.loc.gov/2016030692 1—16 Student Edition ISBN13: 978-0-321-97710-6 www.pearsonhighered.com Student Edition ISBN10: 0-321-97710-6 A01_NAGL7106_07_SE_FM_i-xx.indd 2 31/10/16 1:21 PM Dedicated to R. Kent Nagle He has left his imprint not only on these pages but upon all who knew him. He was that rare mathematician who could effectively communicate at all levels, imparting his love for the subject with the same ease to undergraduates, graduates, precollege students, public school teachers, and his colleagues at the University of South Florida. Kent was at peace in life—a peace that emanated from the depth of his under- standing of the human condition and the strength of his beliefs in the institutions of family, religion, and education. He was a research mathematician, an accomplished author, a Sunday school teacher, and a devoted husband and father. Kent was also my dear friend and my jogging partner who has left me behind still struggling to keep pace with his high ideals. E. B. Saff A01_NAGL7106_07_SE_FM_i-xx.indd 3 31/10/16 1:21 PM
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