INSTRUCTOR’S
M
SOLUTIONS MANUAL
PR
D IFFERENTIAL E QUATIONS
& L INEAR A LGEBRA
ES
FOURTH EDITION
SI
VE
C. Henry Edwards
David E. Penney
G
The University of Georgia
David T. Calvis
R
Baldwin Wallace University
AD
ES
,M
PR
ES
SI
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
VE
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 © 2018, 2010, 2005 Pearson Education, Inc.
Publishing as Pearson, 330 Hudson Street, NY NY 10013
G
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form
or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the
R
publisher. Printed in the United States of America.
AD
ES
ISBN-13: 978-0-13-449825-6
ISBN-10: 0-13-449825-9
, CONTENTS
M
1 FIRST-ORDER DIFFERENTIAL EQUATIONS
1.1 Differential Equations and Mathematical Models 1
PR
1.2 Integrals as General and Particular Solutions 8
1.3 Slope Fields and Solution Curves 16
1.4 Separable Equations and Applications 28
1.5 Linear First-Order Equations 44
ES
1.6 Substitution Methods and Exact Equations 62
Chapter 1 Review Problems 86
2 MATHEMATICAL MODELS
SI
AND NUMERICAL METHODS
2.1 Population Models 101
2.2 Equilibrium Solutions and Stability 117
VE
2.3 Acceleration-Velocity Models 128
2.4 Numerical Approximation: Euler's Method 138
2.5 A Closer Look at the Euler Method 146
2.6 The Runge-Kutta Method 158
G
3 LINEAR SYSTEMS AND MATRICES
R
3.1 Introduction to Linear Systems 173
3.2 Matrices and Gaussian Elimination 177
AD
3.3 Reduced Row-Echelon Matrices 183
3.4 Matrix Operations 192
3.5 Inverses of Matrices 199
3.6 Determinants 208
ES
3.7 Linear Equations and Curve Fitting 219
iii
Copyright © 2018 Pearson Education, Inc.
, 4 VECTOR SPACES
4.1 The Vector Space R 3 229
M
4.2 The Vector Space R n and Subspaces 235
4.3 Linear Combinations and Independence of Vectors 241
PR
4.4 Bases and Dimension for Vector Spaces 249
4.5 Row and Column Spaces 256
4.6 Orthogonal Vectors in R n 262
4.7 General Vector Spaces 268
ES
5 HIGHER-ORDER LINEAR
DIFFERENTIAL EQUATIONS
5.1 Introduction: Second-Order Linear Equations 275
SI
5.2 General Solutions of Linear Equations 282
5.3 Homogeneous Equations with Constant Coefficients 290
5.4 Mechanical Vibrations 298
VE
5.5 Nonhomogeneous Equations and Undetermined Coefficients 309
5.6 Forced Oscillations and Resonance 322
6 EIGENVALUES AND EIGENVECTORS
G
6.1 Introduction to Eigenvalues 335
6.2 Diagonalization of Matrices 349
R
6.3 Applications Involving Powers of Matrices 361
AD
7 LINEAR SYSTEMS OF
DIFFERENTIAL EQUATIONS
7.1 First-Order Systems and Applications 379
7.2 Matrices and Linear Systems 388
7.3 The Eigenvalue Method for Linear Systems 395
ES
7.4 A Gallery of Solution Curves of Linear Systems 427
7.5 Second-Order Systems and Mechanical Applications 433
iv
Copyright © 2018 Pearson Education, Inc.
M
SOLUTIONS MANUAL
PR
D IFFERENTIAL E QUATIONS
& L INEAR A LGEBRA
ES
FOURTH EDITION
SI
VE
C. Henry Edwards
David E. Penney
G
The University of Georgia
David T. Calvis
R
Baldwin Wallace University
AD
ES
,M
PR
ES
SI
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
VE
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 © 2018, 2010, 2005 Pearson Education, Inc.
Publishing as Pearson, 330 Hudson Street, NY NY 10013
G
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form
or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the
R
publisher. Printed in the United States of America.
AD
ES
ISBN-13: 978-0-13-449825-6
ISBN-10: 0-13-449825-9
, CONTENTS
M
1 FIRST-ORDER DIFFERENTIAL EQUATIONS
1.1 Differential Equations and Mathematical Models 1
PR
1.2 Integrals as General and Particular Solutions 8
1.3 Slope Fields and Solution Curves 16
1.4 Separable Equations and Applications 28
1.5 Linear First-Order Equations 44
ES
1.6 Substitution Methods and Exact Equations 62
Chapter 1 Review Problems 86
2 MATHEMATICAL MODELS
SI
AND NUMERICAL METHODS
2.1 Population Models 101
2.2 Equilibrium Solutions and Stability 117
VE
2.3 Acceleration-Velocity Models 128
2.4 Numerical Approximation: Euler's Method 138
2.5 A Closer Look at the Euler Method 146
2.6 The Runge-Kutta Method 158
G
3 LINEAR SYSTEMS AND MATRICES
R
3.1 Introduction to Linear Systems 173
3.2 Matrices and Gaussian Elimination 177
AD
3.3 Reduced Row-Echelon Matrices 183
3.4 Matrix Operations 192
3.5 Inverses of Matrices 199
3.6 Determinants 208
ES
3.7 Linear Equations and Curve Fitting 219
iii
Copyright © 2018 Pearson Education, Inc.
, 4 VECTOR SPACES
4.1 The Vector Space R 3 229
M
4.2 The Vector Space R n and Subspaces 235
4.3 Linear Combinations and Independence of Vectors 241
PR
4.4 Bases and Dimension for Vector Spaces 249
4.5 Row and Column Spaces 256
4.6 Orthogonal Vectors in R n 262
4.7 General Vector Spaces 268
ES
5 HIGHER-ORDER LINEAR
DIFFERENTIAL EQUATIONS
5.1 Introduction: Second-Order Linear Equations 275
SI
5.2 General Solutions of Linear Equations 282
5.3 Homogeneous Equations with Constant Coefficients 290
5.4 Mechanical Vibrations 298
VE
5.5 Nonhomogeneous Equations and Undetermined Coefficients 309
5.6 Forced Oscillations and Resonance 322
6 EIGENVALUES AND EIGENVECTORS
G
6.1 Introduction to Eigenvalues 335
6.2 Diagonalization of Matrices 349
R
6.3 Applications Involving Powers of Matrices 361
AD
7 LINEAR SYSTEMS OF
DIFFERENTIAL EQUATIONS
7.1 First-Order Systems and Applications 379
7.2 Matrices and Linear Systems 388
7.3 The Eigenvalue Method for Linear Systems 395
ES
7.4 A Gallery of Solution Curves of Linear Systems 427
7.5 Second-Order Systems and Mechanical Applications 433
iv
Copyright © 2018 Pearson Education, Inc.
