A First Course in
DIFFERENTIAL EQUATIONS
with Modeling Applications
11E
Dennis G. Zill
,11E
A First Course in Differential Equations
with Modeling Applications
Dennis G. Zill
Loyola Marymount University
Australia ● Brazil ● Mexico ● Singapore ● United Kingdom ● United States
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203
, A First Course in Differential Equations with © 2018, 2013 Cengage Learning
Modeling Applications, Eleventh Edition
ALL RIGHTS RESERVED. No part of this work covered by the copyright herein
Dennis G. Zill may be reproduced or distributed in any form or by any means, except as
Product Director: Terry Boyle permitted by U.S. copyright law, without the prior written permission of the
copyright owner.
Product Manager: Kate Schrumpf
Content Developer: Spencer Arritt
For product information and technology assistance, contact us at
Product Assistant: Stephen Previte
Cengage Learning Customer & Sales Support, 1-800-354-9706.
Marketing Manager: Ana Albinson For permission to use material from this text or product,
Content Project Manager: Jennifer Risden submit all requests online at www.cengage.com/permissions.
Further permissions questions can be e-mailed to
Art Director: Vernon Boes
.
Manufacturing Planner: Becky Cross
Production Service: MPS Limited
Library of Congress Control Number: 2016936077
Photo Researcher: Lumina Datamatics
Student Edition:
Text Researcher: Lumina Datamatics
Copy Editor: Steven Summerlight ISBN: 978-1-305-96572-0
Text Designer: Terri Wright Loose-leaf Edition:
Cover Designer: Terri Wright ISBN: 978-1-337-29312-9
Cover Image: STS-114 Crew, ISS Expedition
11 Crew, NASA Cengage Learning
Compositor: MPS Limited 20 Channel Center Street
Boston, MA 02210
Portions of this text appear in Advanced USA
Engineering Mathematics, Sixth Edition,
Copyright 2018, Jones & Bartlett Learning, Cengage Learning is a leading provider of customized learning solutions
Burlington, MA 01803 and are used with the with employees residing in nearly 40 different countries and sales in more
permission of the publisher. than 125 countries around the world. Find your local representative at
www.cengage.com.
Cengage Learning products are represented in Canada by Nelson Education, Ltd.
To learn more about Cengage Learning Solutions, visit www.cengage.com.
Purchase any of our products at your local college store or at our preferred
online store www.cengagebrain.com.
Printed in the United States of America
Print Number: 01 Print Year: 2016
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203
, Contents
Preface vii
1 Introduction to Differential Equations 2
Kevin George/Shutterstock.com
1.1 Definitions and Terminology 3
1.2 Initial-Value Problems 15
1.3 Differential Equations as Mathematical Models 22
CHAPTER IN RE VIE W
2 First-Order Differential Equations 36
© Joggie Botma/Shutterstock
2.1 Solution Curves Without a Solution 37
2.1.1 Direction Fields 37
2.1.2 Autonomous First-Order DEs 39
2.2 Separable Equations 47
.com
2.3 Linear Equations 55
2.4 Exact Equations 64
2.5 Solutions by Substitutions 72
2.6 A Numerical Method 76
CHAPTER IN RE VIE W
3 Modeling with First-Order Differential
Fotos593/Shutterstock.com
Equations 84
3.1 Linear Models 85
3.2 Nonlinear Models 96
3.3 Modeling with Systems of First-Order DEs 107
CHAPTER IN RE VIE W
4 Higher-Order Differential Equations 118
4.1 Preliminary Theory—Linear Equations 119
4.1.1 Initial-Value and Boundary-Value Problems 119
Bill Ingalls/NASA
4.1.2 Homogeneous Equations 121
4.1.3 Nonhomogeneous Equations 127
4.2 Reduction of Order 132
4.3 Homogeneous Linear Equations with Constant
Coefficients 135
4.4 Undetermined Coefficients—Superposition Approach 142
4.5 Undetermined Coefficients—Annihilator Approach 152
4.6 Variation of Parameters 159
4.7 Cauchy-Euler Equations 166
iii
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203
DIFFERENTIAL EQUATIONS
with Modeling Applications
11E
Dennis G. Zill
,11E
A First Course in Differential Equations
with Modeling Applications
Dennis G. Zill
Loyola Marymount University
Australia ● Brazil ● Mexico ● Singapore ● United Kingdom ● United States
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203
, A First Course in Differential Equations with © 2018, 2013 Cengage Learning
Modeling Applications, Eleventh Edition
ALL RIGHTS RESERVED. No part of this work covered by the copyright herein
Dennis G. Zill may be reproduced or distributed in any form or by any means, except as
Product Director: Terry Boyle permitted by U.S. copyright law, without the prior written permission of the
copyright owner.
Product Manager: Kate Schrumpf
Content Developer: Spencer Arritt
For product information and technology assistance, contact us at
Product Assistant: Stephen Previte
Cengage Learning Customer & Sales Support, 1-800-354-9706.
Marketing Manager: Ana Albinson For permission to use material from this text or product,
Content Project Manager: Jennifer Risden submit all requests online at www.cengage.com/permissions.
Further permissions questions can be e-mailed to
Art Director: Vernon Boes
.
Manufacturing Planner: Becky Cross
Production Service: MPS Limited
Library of Congress Control Number: 2016936077
Photo Researcher: Lumina Datamatics
Student Edition:
Text Researcher: Lumina Datamatics
Copy Editor: Steven Summerlight ISBN: 978-1-305-96572-0
Text Designer: Terri Wright Loose-leaf Edition:
Cover Designer: Terri Wright ISBN: 978-1-337-29312-9
Cover Image: STS-114 Crew, ISS Expedition
11 Crew, NASA Cengage Learning
Compositor: MPS Limited 20 Channel Center Street
Boston, MA 02210
Portions of this text appear in Advanced USA
Engineering Mathematics, Sixth Edition,
Copyright 2018, Jones & Bartlett Learning, Cengage Learning is a leading provider of customized learning solutions
Burlington, MA 01803 and are used with the with employees residing in nearly 40 different countries and sales in more
permission of the publisher. than 125 countries around the world. Find your local representative at
www.cengage.com.
Cengage Learning products are represented in Canada by Nelson Education, Ltd.
To learn more about Cengage Learning Solutions, visit www.cengage.com.
Purchase any of our products at your local college store or at our preferred
online store www.cengagebrain.com.
Printed in the United States of America
Print Number: 01 Print Year: 2016
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203
, Contents
Preface vii
1 Introduction to Differential Equations 2
Kevin George/Shutterstock.com
1.1 Definitions and Terminology 3
1.2 Initial-Value Problems 15
1.3 Differential Equations as Mathematical Models 22
CHAPTER IN RE VIE W
2 First-Order Differential Equations 36
© Joggie Botma/Shutterstock
2.1 Solution Curves Without a Solution 37
2.1.1 Direction Fields 37
2.1.2 Autonomous First-Order DEs 39
2.2 Separable Equations 47
.com
2.3 Linear Equations 55
2.4 Exact Equations 64
2.5 Solutions by Substitutions 72
2.6 A Numerical Method 76
CHAPTER IN RE VIE W
3 Modeling with First-Order Differential
Fotos593/Shutterstock.com
Equations 84
3.1 Linear Models 85
3.2 Nonlinear Models 96
3.3 Modeling with Systems of First-Order DEs 107
CHAPTER IN RE VIE W
4 Higher-Order Differential Equations 118
4.1 Preliminary Theory—Linear Equations 119
4.1.1 Initial-Value and Boundary-Value Problems 119
Bill Ingalls/NASA
4.1.2 Homogeneous Equations 121
4.1.3 Nonhomogeneous Equations 127
4.2 Reduction of Order 132
4.3 Homogeneous Linear Equations with Constant
Coefficients 135
4.4 Undetermined Coefficients—Superposition Approach 142
4.5 Undetermined Coefficients—Annihilator Approach 152
4.6 Variation of Parameters 159
4.7 Cauchy-Euler Equations 166
iii
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203
