Solution Manual for Differential Equations and Linear Algebra 4th Edition by C. Edwards

INSTRUCTOR’S
SOLUTIONS MANUAL

D IFFERENTIAL E QUATIONS
AP
& L INEAR A LGEBRA
FOURTH EDITION
LU

C. Henry Edwards
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David E. Penney
The University of Georgia
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David T. Calvis
Baldwin Wallace University

,CONTENTS


1 FIRST-ORDER DIFFERENTIAL EQUATIONS
1.1 Differential Equations and Mathematical Models 1
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
AP
1.6 Substitution Methods and Exact Equations 62
Chapter 1 Review Problems 86


2 MATHEMATICAL MODELS
AND NUMERICAL METHODS
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2.1 Population Models 101
2.2 Equilibrium Solutions and Stability 117
2.3 Acceleration-Velocity Models 128
2.4 Numerical Approximation: Euler's Method 138
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2.5 A Closer Look at the Euler Method 146
2.6 The Runge-Kutta Method 158


3 LINEAR SYSTEMS AND MATRICES
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3.1 Introduction to Linear Systems 173
3.2 Matrices and Gaussian Elimination 177
3.3 Reduced Row-Echelon Matrices 183
3.4 Matrix Operations 192
3.5 Inverses of Matrices 199
3.6 Determinants 208
3.7 Linear Equations and Curve Fitting 219




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,4 VECTOR SPACES
4.1 The Vector Space R 3 229
4.2 The Vector Space R n and Subspaces 235
4.3 Linear Combinations and Independence of Vectors 241
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
AP
5 HIGHER-ORDER LINEAR
DIFFERENTIAL EQUATIONS
5.1 Introduction: Second-Order Linear Equations 275
5.2 General Solutions of Linear Equations 282
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5.3 Homogeneous Equations with Constant Coefficients 290
5.4 Mechanical Vibrations 298
5.5 Nonhomogeneous Equations and Undetermined Coefficients 309
5.6 Forced Oscillations and Resonance 322
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6 EIGENVALUES AND EIGENVECTORS
6.1 Introduction to Eigenvalues 335
6.2 Diagonalization of Matrices 349
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6.3 Applications Involving Powers of Matrices 361


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
7.4 A Gallery of Solution Curves of Linear Systems 427
7.5 Second-Order Systems and Mechanical Applications 433

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, 7.6 Multiple Eigenvalue Solutions 445
7.7 Numerical Methods for Systems 464


8 MATRIX EXPONENTIAL METHODS
8.1 Matrix Exponentials and Linear Systems 473
8.2 Nonhomogeneous Linear Systems 483
8.3 Spectral Decomposition Methods 491


9 NONLINEAR SYSTEMS AND PHENOMENA
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9.1 Stability and the Phase Plane 511
9.2 Linear and Almost Linear Systems 520
9.3 Ecological Applications: Predators and Competitors 538
9.4 Nonlinear Mechanical Systems 553
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10 LAPLACE TRANSFORM METHODS
10.1 Laplace Transforms and Inverse Transforms 565
10.2 Transformation of Initial Value Problems 570
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10.3 Translation and Partial Fractions 579
10.4 Derivatives, Integrals, and Products of Transforms 588
10.5 Periodic and Piecewise Continuous Input Functions 595
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11 POWER SERIES METHODS
11.1 Introduction and Review of Power Series 609
11.2 Power Series Solutions 615
11.3 Frobenius Series Solutions 628
11.4 Bessel Functions 642


APPENDIX A
Existence and Uniqueness of Solutions 649



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