net http://www.net LIBROS UNIVERISTARIOS Y SOLUCIONARIOS DE MUCHOS DE ESTOS LIBROS LOS SOLUCIONARIOS CONTIENEN TODOS LOS EJERCICIOS DEL LIBRO RESUELTOS Y EXPLICADOS DE FORMA CLARA VISITANOS PARA DESARGALOS GRATIS.net REVIEW OF DIFFERENTIATION a a a a a www.net Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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Tenth Edition A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications DENNIS G. ZILL Loyola Marymount University Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States www.net Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
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a Macmillan Company Instructors: Please visit login.com and log in to access instructor-specific resource .8 of this text appears in Advanced Engineering Mathematics, Fourth Edition, Copyright 2011, Jones & Bartlett Learning, Burlington, MA 01803 and is used with the permission of the publisher. Printed in the United States of America 1 2 3 4 5 6 7 16 15 14 13 12 www.net Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.net Contents Preface ix Projects P-1 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1 1.1 Definitions and Terminology 2 1.2 Initial-Value Problems 13 1.3 Differential Equations as Mathematical Models 20 Chapter 1 in Review 33 2 FIRST-ORDER DIFFERENTIAL EQUATIONS 35 2.1 Solution Curves Without a Solution 36 2.2 Autonomous First-Order DEs 38 2.5 Solutions by Substitutions 71 2.6 A Numerical Method 75 Chapter 2 in Review 80 3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 83 3.3 Modeling with Systems of First-Order DEs 106 Chapter 3 in Review 113 v www.net Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. vi ● CONTENTS 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS 116 4.1 Preliminary Theory—Linear Equations 117 4.1 Initial-Value and Boundary-Value Problems 117 4.2 Reduction of Order 129 4.3 Homogeneous Linear Equations with Constant Coefficient 132 4.4 Undetermined Coefficients—Superposition Approach 139 4.5 Undetermined Coefficients—Annihilator Approach 149 4.6 Variation of Parameters 156 4.7 Cauchy-Euler Equation 162 4.1 Initial-Value Problems 169 4.2 Boundary-Value Problems 176 4.9 Solving Systems of Linear DEs by Elimination 180 4.10 Nonlinear Differential Equations 185 Chapter 4 in Review 190 5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 192 5.1 Linear Models: Initial-Value Problems 193 5.1 Spring/Mass Systems: Free Undamped Motion 193 5.2 Spring/Mass Systems: Free Damped Motion 197 5.3 Spring/Mass Systems: Driven Motion 200 5.4 Series Circuit Analogue 203 5.2 Linear Models: Boundary-Value Problems 210 5.3 Nonlinear Models 218 Chapter 5 in Review 228 6 SERIES SOLUTIONS OF LINEAR EQUATIONS 231 6.1 Review of Power Series 232 6.2 Solutions About Ordinary Points 238 6.3 Solutions About Singular Points 247 6.4 Special Functions 257 Chapter 6 in Review 271 www.net Copyright 2012 Cengage Learning. All Rights Reserved.
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CONTENTS ● vii 7 THE LAPLACE TRANSFORM 273 7.1 Definition of the Laplace Transform 274 7.2 Inverse Transforms and Transforms of Derivatives 281 7.2 Transforms of Derivatives 284 7.1 Translation on the s-Axis 290 7.2 Translation on the t-Axis 293 7.4 Operational Properties II 301 7.1 Derivatives of a Transform 301 7.2 Transforms of Integrals 302 7.3 Transform of a Periodic Function 307 7.5 The Dirac Delta Function 312 7.6 Systems of Linear Differential Equations 315 Chapter 7 in Review 320 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 325 8.1 Preliminary Theory—Linear Systems 326 8.2 Homogeneous Linear Systems 333 8.1 Distinct Real Eigenvalues 334 8.3 Nonhomogeneous Linear Systems 348 8.2 Variation of Parameters 351 8.4 Matrix Exponential 356 Chapter 8 in Review 360 9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 362 9.1 Euler Methods and Error Analysis 363 9.2 Runge-Kutta Methods 368 9.4 Higher-Order Equations and Systems 375 9.5 Second-Order Boundary-Value Problems 380 Chapter 9 in Review 384 www.net Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. viii ● CONTENTS APPENDIXES I Gamma Function APP-1 II Matrices APP-3 III Laplace Transforms APP-21 Answers for Selected Odd-Numbered Problems ANS-1 Index I-1 www.net Copyright 2012 Cengage Learning. All Rights Reserved.
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Preface TO THE STUDENT Authors of books live with the hope that someone actually reads them. Contrary to what you might believe, almost everything in a typical college-level mathematics text is written for you, and not the instructor. True, the topics covered in the text are chosen to appeal to instructors because they make the decision on whether to use it in their classes, but everything written in it is aimed directly at you, the student. So I want to encourage you—no, actually I want to tell you—to read this textbook! But do not read this text like you would a novel; you should not read it fast and you should not skip anything.
Think of it as a workbook. By this I mean that mathemat- ics should always be read with pencil and paper at the ready because, most likely, you will have to work your way through the examples and the discussion.