VE
R
INSTRUCTOR’S
IF
SOLUTIONS MANUAL
IE
D
BR
AI
DIFFERENTIAL EQUATIONS
N
BO
AND BOUNDARY VALUE PROBLEMS
O
COMPUTING AND MODELING
ST
SIXTH EDITION
ER
C. Henry Edwards
David E. Penney
The University of Georgia
David Calvis
Baldwin-Wallace University
Boston Columbus Indianapolis New York San Francisco Hoboken
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
VERIFIEDBRAINBOOSTER
, VE
R
IF
IE
D
BR
AI
N
BO
O
ST
ER
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2023, 2019, 2015 by Pearson Education, Inc. or its affiliates. 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 and Permissions
Department, please visit www.pearsoned.com/permissions.
ISBN-13: 978-0-13-754027-3
ISBN-10: 0-13-754027-2
1 2 3 4 5 6 online 17 16 15 14
www.pearsonhighered.com
VERIFIEDBRAINBOOSTER
, VE
Contents
R
1 First-Order Differential Equations
1.1 Differential Equations and Mathematical Models 1
IF
1.2 Integrals as General and Particular Solutions 8
IE
1.3 Slope Fields and Solution Curves 16
1.4 Separable Equations and Applications 27
D
1.5 Linear First-Order Equations 44
BR
1.6 Substitution Methods and Exact Equations 62
Chapter 1 Review Problems 86
AI
2 Mathematical Models and Numerical Methods
2.1 Population Models 100
N
2.2 Equilibrium Solutions and Stability 116
BO
2.3 Acceleration-Velocity Models 127
2.4 Numerical Approximation: Euler's Method 137
2.5 A Closer Look at the Euler Method 144
O
2.6 The Runge-Kutta Method 155
ST
3 Linear Equations of Higher Order
3.1 Introduction: Second-Order Linear Equations 167
ER
3.2 General Solutions of Linear Equations 174
3.3 Homogeneous Equations with Constant Coefficients 182
3.4 Mechanical Vibrations 190
3.5 Nonhomogeneous Equations and Undetermined Coefficients 201
3.6 Forced Oscillations and Resonance 214
3.7 Electrical Circuits 227
3.8 Endpoint Problems and Eigenvalues 234
4 Introduction to Systems of Differential Equations
4.1 First-Order Systems and Applications 241
4.2 The Method of Elimination 250
4.3 Numerical Methods for Systems 270
5 Linear Systems of Differential Equations
5.1 Matrices and Linear Systems 280
5.2 The Eigenvalue Method for Homogeneous Linear Systems 288
5.3 Solution Curves of Linear Systems 313
5.4 Second-Order Systems and Mechanical Applications 319
5.5 Multiple Eigenvalue Solutions 331
5.6 Matrix Exponentials and Linear Systems 345
5.7 Nonhomogeneous Linear Systems 355
iii
Copyright © 2023 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
6 Nonlinear Systems and Phenomena
6.1 Stability and the Phase Plane 363
R
6.2 Linear and Almost Linear Systems 372
6.3 Ecological Applications: Predators and Competitors 389
IF
6.4 Nonlinear Mechanical Systems 404
IE
6.5 Chaos in Dynamical Systems 415
D
7 Laplace Transform Methods
7.1 Laplace Transforms and Inverse Transforms 422
BR
7.2 Transformation of Initial Value Problems 427
7.3 Translation and Partial Fractions 436
AI
7.4 Derivatives, Integrals, and Products of Transforms 444
7.5 Periodic and Piecewise Continuous Input Functions 451
N
7.6 Impulses and Delta Functions 464
BO
8 Power Series Methods
8.1 Introduction and Review of Power Series 473
O
8.2 Series Solutions Near Ordinary Points 479
8.3 Regular Singular Points 492
ST
8.4 Method of Frobenius—The Exceptional Cases 505
8.5 Bessel’s Equation 514
ER
8.6 Applications of Bessel Functions 522
9 Fourier Series Methods and Partial Differential Equations
9.1 Periodic Functions and Trigonometric Series 527
9.2 General Fourier Series and Convergence 537
9.3 Fourier Sine and Cosine Series 551
9.4 Applications of Fourier Series 565
9.5 Heat Conduction and Separation of Variables 571
9.6 Vibrating Strings and the One-Dimensional Wave Equation 577
9.7 Steady-State Temperature and Laplace’s Equation 585
10 Eigenvalue Methods and Boundary Value Problems
10.1 Sturm-Liouville Problems and Eigenfunction Expansions 596
10.2 Applications of Eigenfunction Series 607
10.3 Steady Periodic Solutions and Natural Frequencies 619
10.4 Cylindrical Coordinate Problems 631
10.5 Higher-Dimensional Phenomena 643
Appendix
Existence and Uniqueness of Solutions 644
iv
Copyright © 2023 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
R
INSTRUCTOR’S
IF
SOLUTIONS MANUAL
IE
D
BR
AI
DIFFERENTIAL EQUATIONS
N
BO
AND BOUNDARY VALUE PROBLEMS
O
COMPUTING AND MODELING
ST
SIXTH EDITION
ER
C. Henry Edwards
David E. Penney
The University of Georgia
David Calvis
Baldwin-Wallace University
Boston Columbus Indianapolis New York San Francisco Hoboken
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
VERIFIEDBRAINBOOSTER
, VE
R
IF
IE
D
BR
AI
N
BO
O
ST
ER
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2023, 2019, 2015 by Pearson Education, Inc. or its affiliates. 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 and Permissions
Department, please visit www.pearsoned.com/permissions.
ISBN-13: 978-0-13-754027-3
ISBN-10: 0-13-754027-2
1 2 3 4 5 6 online 17 16 15 14
www.pearsonhighered.com
VERIFIEDBRAINBOOSTER
, VE
Contents
R
1 First-Order Differential Equations
1.1 Differential Equations and Mathematical Models 1
IF
1.2 Integrals as General and Particular Solutions 8
IE
1.3 Slope Fields and Solution Curves 16
1.4 Separable Equations and Applications 27
D
1.5 Linear First-Order Equations 44
BR
1.6 Substitution Methods and Exact Equations 62
Chapter 1 Review Problems 86
AI
2 Mathematical Models and Numerical Methods
2.1 Population Models 100
N
2.2 Equilibrium Solutions and Stability 116
BO
2.3 Acceleration-Velocity Models 127
2.4 Numerical Approximation: Euler's Method 137
2.5 A Closer Look at the Euler Method 144
O
2.6 The Runge-Kutta Method 155
ST
3 Linear Equations of Higher Order
3.1 Introduction: Second-Order Linear Equations 167
ER
3.2 General Solutions of Linear Equations 174
3.3 Homogeneous Equations with Constant Coefficients 182
3.4 Mechanical Vibrations 190
3.5 Nonhomogeneous Equations and Undetermined Coefficients 201
3.6 Forced Oscillations and Resonance 214
3.7 Electrical Circuits 227
3.8 Endpoint Problems and Eigenvalues 234
4 Introduction to Systems of Differential Equations
4.1 First-Order Systems and Applications 241
4.2 The Method of Elimination 250
4.3 Numerical Methods for Systems 270
5 Linear Systems of Differential Equations
5.1 Matrices and Linear Systems 280
5.2 The Eigenvalue Method for Homogeneous Linear Systems 288
5.3 Solution Curves of Linear Systems 313
5.4 Second-Order Systems and Mechanical Applications 319
5.5 Multiple Eigenvalue Solutions 331
5.6 Matrix Exponentials and Linear Systems 345
5.7 Nonhomogeneous Linear Systems 355
iii
Copyright © 2023 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
6 Nonlinear Systems and Phenomena
6.1 Stability and the Phase Plane 363
R
6.2 Linear and Almost Linear Systems 372
6.3 Ecological Applications: Predators and Competitors 389
IF
6.4 Nonlinear Mechanical Systems 404
IE
6.5 Chaos in Dynamical Systems 415
D
7 Laplace Transform Methods
7.1 Laplace Transforms and Inverse Transforms 422
BR
7.2 Transformation of Initial Value Problems 427
7.3 Translation and Partial Fractions 436
AI
7.4 Derivatives, Integrals, and Products of Transforms 444
7.5 Periodic and Piecewise Continuous Input Functions 451
N
7.6 Impulses and Delta Functions 464
BO
8 Power Series Methods
8.1 Introduction and Review of Power Series 473
O
8.2 Series Solutions Near Ordinary Points 479
8.3 Regular Singular Points 492
ST
8.4 Method of Frobenius—The Exceptional Cases 505
8.5 Bessel’s Equation 514
ER
8.6 Applications of Bessel Functions 522
9 Fourier Series Methods and Partial Differential Equations
9.1 Periodic Functions and Trigonometric Series 527
9.2 General Fourier Series and Convergence 537
9.3 Fourier Sine and Cosine Series 551
9.4 Applications of Fourier Series 565
9.5 Heat Conduction and Separation of Variables 571
9.6 Vibrating Strings and the One-Dimensional Wave Equation 577
9.7 Steady-State Temperature and Laplace’s Equation 585
10 Eigenvalue Methods and Boundary Value Problems
10.1 Sturm-Liouville Problems and Eigenfunction Expansions 596
10.2 Applications of Eigenfunction Series 607
10.3 Steady Periodic Solutions and Natural Frequencies 619
10.4 Cylindrical Coordinate Problems 631
10.5 Higher-Dimensional Phenomena 643
Appendix
Existence and Uniqueness of Solutions 644
iv
Copyright © 2023 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
