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Differentiation rules 1. Constant:-C=0 2. Constant Multiple:, cf(r)=cf(x) 3.Sum:,[f(x)±g(x)]=f'(x)±g 4. Product:, f(x)g(x)=f(r)g(x)+ g(r)f(x) d f(r) g(x)f()-f(x)g(x) 5. Quotient [g(x)]2 6. Chain:,f(g())=f(g(x))g(x) d 7. Power 8. Power:lg(x)"=nlg(x)g(x) Derivatives of functions Trigonometric 9 Sin x coS x 10 COSx=-sin x 11.-tanx= secx dx d d 12. cot x=-csc-x 13. sec x sec r tan x csc x cot x x Inverse trigonometric d 15 16.-cos x 17, tanx 1+x 18 cotx 19.-sec x 20. cSc X 1+ Hyperbolic sinh x= cosh x cosh x -tanh x= sech d coth csch x 25.-sech x =-sech x tanh x 26.— schr csch x cothi x Inverse hyperbolic l 27 sinh 28 cosh= 29 tanh x2+1 d 32,csch x x Vx2+1 Exponential 33.—e 34 b"= b(n b Logarithmic. 35 dx logh x x(n Of an integral d d 37 g(t) 8 38 (x, t)dt I Integration Formulas n+1 +C,n≠-1 u= Inu C n e au In b 5. sin udu cosu + c 6. cos u du= sin u+ C 7.sec udu= tan u + C 8.csc2udu cotu+ c 9. sec u tan udu= sec u + C 10. csc u cot udu =-cscu+C 11. tan udu 2. cot u du= In/sin u|+C 13. sec udu= Inlsec u tan ul+ C u du=In/csc u-cotul+ C 15.u sin u du= sin u -u cos u+ C 16. u cosu du cos u +u sin u C 17 inu du u- 4sin 2u C 18.cos2udu=2u+sin 2u+C sin(a- bu sin(a t b)u in(a- b)u sin(a t b)u 19. sin au sin budu 20. cos au cos bu du +o 2(a+b) 2(a-b) 2(a+b) 21.e sin budu (a sin bu- b cos bu)+ C 22. e cos budu a" in(a cos bu t b sin bu)+ C 23. sinh u du= cosh u+C 24. cosh u du sinh u C 25. sechu du= tanh u +C 26. csch2udu=-cothu+C 27. tanh u du= In(cosh u)+C 28. cothu du=In/sinhu+C du =uIn u 30.uIn udu=2u2Inu-iu d u sin 32. du= InJu+ va+ l 33 u' du=-Vv sin + c 34.Va2+ Va+u+Inu+ vatu+ c 1.|a+u 33 du =-tan a- t u du InJu+ vu C Inu+V t C ADVANCED ENGINEERING ATHEMATICS SIXTH EDITION Dennis g. zill Loyola Marymount University J○NES& BARTLETT LEARNING World Headquarters Jones Bartlett Learning 5 Wall Street Burlington, MA 01803 978-443-5000 www.jblearning.coill Jones Bartlett Learning books and products are available through most bookstores and online booksellers. To contact Jones Bartlett Learning directly, call 800-832-0034,fax978-443-8000, Substantial discounts on bulk quantities of Jones Bartlett Leaning publications are available to corporations, professional associations, and other qualified organizations. For details and specific discount information, contact the special sales department at Jones bartlett Learning via the above contact information Copyright 2018 by Jones Bartlett I earning, LI.C, an Ascend L earning Company All rights reserved. No part of the material protected by this copyright may be reproduced or utilized in any form, electronic or mechanical, including photocop ng, recording, or by any information storage and retrieval system, without written permission from the copyright owner The content,statements,views,and opinions herein are the sole expression of the respective authors and not that of Jones&Bartlett Learning, LLCReference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not constitute or imply its endorsement or recommendation by Jones Bartlett Learning, LLC and such reference shall not be used for advertising or product endorsement purposes. All trademarks displayed are the trademarks of the parties noted herein. Advanced Engineering Mathematics, Sixth Edition is an independent publication and has not been au thorized, sponsored, or otherwise approved by the owners of the trademarks or service marks referenced in this product. There may be images in this book that feature models; these models do not necessarily endorse, represent, or participate in the activities represented in the im ages. Any screenshots in this product are for educational and instructive purposes only. Any individuals and scenarios featured in the case studies throughout this ct ma are use Instructional purposes only. Production Credits VP Executive publisher: David D. cella Executive editor: matt Kane Acquisitions Editor: Laura Pagluica Associate Editor: Taylor Ferracane Vendor Manager: Sara Kelly Director of marketing: Andrea dcfronzo VP Manufacturing and Inventory control Therese Connell Composition and Project Management: Aptara, Inc Cover Design: Kristin E. Parker Rights media Specialist: Merideth tumasz Media Development Editor: Shannon Sheehan Cover Images: Domestic: O NASA International: C CHEN MIN CHUN/Shutterstock Printing and Binding: RR Donnelley Cover Printing: RR Donnelley To order this product, use ISBN: 978-1-284-10590-2 Library of Congress Cataloging-in-Publication Data Author: Zill. Dennis g Title: Advanced Engineering Mathematics/Dennis G. Zill, Loyola Marymount University Description: Sixth edition. I Burlington, MA: Jones Bartlett Learning, 「2017| ncludes index Identifiers: LCCN 20160224101 ISBN 9781284105902(casebound)I ISBN 1284105903(casebound) Subjects: LCSH: Engineering mathematics Classification: LCC TA330.Z55 2017 DDC 620.001/51-dc23 Lcrecordavailableat 6048 Printed in the Lnited States of america 201918171610987654321 Contents 38e Preface PART Ordinary Differential Equations 1 Introduction to Diferential Equations 1.1 Definitions and Terminology 12 Initial-Value problems 14 1.3 Differential equations as mathematical models Chapter 1 in reviev First-Order Differential Equations 33 2.1 Solution Curves Without a solution 34 2.1.1 Direction Fields 34 2.1.2 Autonomous first-Order des 2.2 Separable Equations 43 2.3 Linear equations 50 2.4 Exact Equations 59 2.5 Solutions by substitutions 2.6 A Numerical Method 69 2.7 Linear models 74 2. 8 Nonlinear models 84 2. 9 Modeling with Systems of First-Order dEs Chapter 2 in reviet 99 3 Higher-Order Differential Equations 105 3.1 Theory of linear equations 3.1.1 Initial-Value and boundary-Value Problems 106 3.1.2 Homogeneous equations 3.1.3 Nonhomogeneous equations 113 A38 3.2 Reduction of order 117 3.3 Homogeneous Linear Equations with Constant Coefficients 120 3.4 Undetermined coefficients 127 3.5 Variation of parameters 136 3.6 Cauchy-Euler Equations 141 3.7 Nonlinear Equations 147 3.8 Linear models: Initial-Value Problems 15 3.8.1 Spring/Mass Systems: Free Undamped Motion 152 3.8.2 Spring/Mass Systems: Free Damped Motion 155 3.8.3 Spring/Mass Systems: Driven Motion 158 3.8.4 Series Circuit analogue 161 3.9 Linear Models: Boundary-Value Problems 167 3.10 Green s functions 177 3.10.1 nitial-Value Problems 3.10.2 Boundary-Value Problems 183 3.11 Nonlinear models 187 3.12 Solving Systems of Linear Equations Chapter 3 in Review 203 The Laplace Transform 211 4.1 Definition of the Laplace Transform 212 4.2 The Inverse transform and transforms of derivatives 218 4.2.1 Inverse transforms 218 4. 2.2 Transforms of derivatives 220 4.3 Translation Theorems 226 4.3.1 Translation on the s-axis 226 432 Translation on the t-axis 229 4.4 Additional Operational Properties 236 4.4.1 Derivatives of transforms 4.4.2 Transforms of Integrals 238 4.4.3 Transform of a periodic function 244 4.5 The dirac delta function 248 4.6 Systems of Linear Differential equations 251 Chapter 4 in review 257 iy Contents 5 Series Solutions of Linear Differential Equations 261 5.1 Solutions about Ordinary Poin 262 5.1.1 Review of power series 262 5.1.2 Power Series solutions 264 5.2 Solutions about singular points 271 5.3 Special Functions 280 5.3.1 Bessel functions 280 5.3.2 Legendre Functions 288 Chapter 5 in Review 294 6 Numerical Solutions of Ordinary Differential Equations 297 6.1 Euler Methods and Error analysis 298 6.2 Runge-Kutta Methods 302 6.3 Multistep Methods 307 6.4 Higher-Order Equations and Systems 309 6.5 Second-Order Boundary-Value Problems 313 Chapter 6 Rt in Review 317 PART 2 Vectors, Matrices, and vector calculus 319 vEctors 321 7.1 lectors in 2-Space 322 7.2 Vectors in 3-Space 327 7.3 Dot Product 332 7.4 Cross product 338 7.5 Lines and planes in 3-space 345 7.6 Vector Spaces 351 7.7 Gram-Schmidt Orthogonalization Process 359 Chapter 7 in Review 364 Contents v

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