Application Modules vi
Preface ix
Chapter 1 First-Order Differential Equations 1
1.1 Differential Equations and Mathematical Models 1
1.2 Integrals as General and Particular Solutions 10
1.3 Slope Fields and Solution Curves 17
1.4 Separable Equations and Applications 30
1.5 Linear First-Order Equations 46
1.6 Substitution Methods and Exact Equations 58
Chapter 2 Mathematical Models and Numerical Methods 75
2.1 Population Models 75
2.2 Equilibrium Solutions and Stability 87
2.3 Acceleration-Velocity Models 94
2.4 Numerical Approximation: Euler's Method 106
2.5 A Closer Look at the Euler Method 117
2.6 The Runge-Kutta Method 127
Chapter 3 Linear Systems and Matrices 138
3.1 Introduction to Linear Systems 147
3.2 Matrices and Gaussian Elimination 146
3.3 Reduced Row-Echelon Matrices 156
3.4 Matrix Operations 164
3.5 Inverses of Matrices 175
3.6 Determinants 188
3.7 Linear Equations and Curve Fitting 203
Chapter 4 Vector Spaces 211
4.1 The Vector Space R3R3 211
4.2 The Vector Space RnRn 221
4.3 Linear Combinations and Independence of Vectors 228
4.4 Bases and Dimension for Vector Spaces 235
4.5 Row and Column Spaces 242
4.6 Orthogonal Vectors in RnRn 250
4.7 General Vector Spaces 257
Chapter 5 Higher-Order Linear Differential Equations 265
5.1 Introduction: Second-Order Linear Equations 265
5.2 General Solutions of Linear Equations 279
5.3 Homogeneous Equations with Constant Coefficients 291
5.4 Mechanical Vibrations 302
5.5 Nonhomogeneous Equations and Undetermined Coefficients 314
5.6 Forced Oscillations and Resonance 327
Chapter 6 Eigenvalues and Eigenvectors 339
6.1 Introduction to Eigenvalues 339
6.2 Diagonalization of Matrices 347
6.3 Applications Involving Powers of Matrices 354
Chapter 7 Linear Systems of Differential Equations 365
7.1 First-Order Systems and Applications 365
7.2 Matrices and Linear Systems 375
7.3 The Eigenvalue Method for Linear Systems 385
7.4 A Gallery of Solution Curves of Linear Systems 398
7.5 Second-Order Systems and Mechanical Applications 424
7.6 Multiple Eigenvalue Solutions 437
7.7 Numerical Methods for Systems 454
Chapter 8 Matrix Exponential Methods 469
8.1 Matrix Exponentials and Linear Systems 469
8.2 Nonhomogeneous Linear Systems 482
8.3 Spectral Decomposition Methods 490
Chapter 9 Nonlinear Systems and Phenomena 503
9.1 Stability and the Phase Plane 503
9.2 Linear and Almost Linear Systems 514
9.3 Ecological Models: Predators and Competitors 526
9.4 Nonlinear Mechanical Systems 539
Chapter 10 Laplace Transform Methods 557
10.1 Laplace Transforms and Inverse Transforms 557
10.2 Transformation of Initial Value Problems 567
10.3 Translation and Partial Fractions 578
10.4 Derivatives, Integrals, and Products of Transforms 587
10.5 Periodic and Piecewise Continuous Input Functions 594
Chapter 11 Power Series Methods 604
11.1 Introduction and Review of Power Series 604
11.2 Power Series Solutions 616
11.3 Frobenius Series Solutions 627
11.4 Bessel Functions 642
References for Further Study 652
Appendix A: Existence and Uniqueness of Solutions 654
Appendix B: Theory of Determinants 668
Answers to Selected Problems 677
Index 733
18.119.29.105