SOLUTION MANUAL FOR
DIFFERENTIAL EQUATIONS AND
LINEAR ALGEBRA, 4TH EDITION
STEPHEN GOODE, SCOTT ANNIN
,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
1.6 Substitution Methods and Exact Equations 62
Chapter 1 Review Problems 86
2 MATHEMATICAL MODELS
AND NUMERICAL METHODS
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
2.5 A Closer Look at the Euler Method 146
2.6 The Runge-Kutta Method 158
3 LINEAR SYSTEMS AND MATRICES
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
iii
,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
5 HIGHER-ORDER LINEAR
DIFFERENTIAL EQUATIONS
5.1 Introduction: Second-Order Linear Equations 275
5.2 General Solutions of Linear Equations 282
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
6 EIGENVALUES AND EIGENVECTORS
6.1 Introduction to Eigenvalues 335
6.2 Diagonalization of Matrices 349
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
iv
, 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
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
10 LAPLACE TRANSFORM METHODS
10.1 Laplace Transforms and Inverse Transforms 565
10.2 Transformation of Initial Value Problems 570
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
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
v
DIFFERENTIAL EQUATIONS AND
LINEAR ALGEBRA, 4TH EDITION
STEPHEN GOODE, SCOTT ANNIN
,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
1.6 Substitution Methods and Exact Equations 62
Chapter 1 Review Problems 86
2 MATHEMATICAL MODELS
AND NUMERICAL METHODS
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
2.5 A Closer Look at the Euler Method 146
2.6 The Runge-Kutta Method 158
3 LINEAR SYSTEMS AND MATRICES
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
iii
,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
5 HIGHER-ORDER LINEAR
DIFFERENTIAL EQUATIONS
5.1 Introduction: Second-Order Linear Equations 275
5.2 General Solutions of Linear Equations 282
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
6 EIGENVALUES AND EIGENVECTORS
6.1 Introduction to Eigenvalues 335
6.2 Diagonalization of Matrices 349
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
iv
, 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
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
10 LAPLACE TRANSFORM METHODS
10.1 Laplace Transforms and Inverse Transforms 565
10.2 Transformation of Initial Value Problems 570
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
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
v
