CHAPTER 1. First-Order Differential Equations 1.1 Differential Equations and Mathematical Models
1.2 Integrals as General and Particular Solutions
1.3 Slope Fields and Solution Curves
1.4 Separable Equations and Applications
1.5 Linear First-Order Equations
1.6 Substitution Methods and Exact Equations
CHAPTER 2. Mathematical Models and Numerical Methods
2.1 Population Models
2.2 Equilibrium Solutions and Stability
2.3 Acceleration–Velocity Models
2.4 Numerical Approximation: Euler's Method
2.5 A Closer Look at the Euler Method
2.6 The Runge–Kutta Method
CHAPTER 3. Linear Systems and Matrices
3.1 Introduction to Linear Systems
3.2 Matrices and Gaussian Elimination
3.3 Reduced Row-Echelon Matrices
3.4 Matrix Operations
3.5 Inverses of Matrices
3.6 Determinants
3.7 Linear Equations and Curve Fitting
CHAPTER 4. Vector Spaces
4.1 The Vector Space R3
4.2 The Vector Space Rn and Subspaces
4.3 Linear Combinations and Independence of Vectors
4.4 Bases and Dimension for Vector Spaces
4.5 Row and Column Spaces
4.6 Orthogonal Vectors in Rn
4.7 General Vector Spaces
CHAPTER 5. Higher-Order Linear Differential Equations
5.1 Introduction: Second-Order Linear Equations
5.2 General Solutions of Linear Equations
5.3 Homogeneous Equations with Constant Coefficients
5.4 Mechanical Vibrations
5.5 Nonhomogeneous Equations and Undetermined Coefficients
5.6 Forced Oscillations and Resonance
CHAPTER 6. Eigenvalues and Eigenvectors
6.1 Introduction to Eigenvalues
6.2 Diagonalization of Matrices
6.3 Applications Involving Powers of Matrices
CHAPTER 7. Linear Systems of Differential Equations
7.1 First-Order Systems and Applications
7.2 Matrices and Linear Systems
7.3 The Eigenvalue Method for Linear Systems
7.4 Second-Order Systems and Mechanical Applications
7.5 Multiple Eigenvalue Solutions
7.6 Numerical Methods for Systems
CHAPTER 8. Matrix Exponential Methods
8.1 Matrix Exponentials and Linear Systems
8.2 Nonhomogeneous Linear Systems
8.3 Spectral Decomposition Methods
CHAPTER 9. Nonlinear Systems and Phenomena
9.1 Stability and the Phase Plane
9.2 Linear and Almost Linear Systems
9.3 Ecological Models: Predators and Competitors
9.4 Nonlinear Mechanical Systems
CHAPTER 10. Laplace Transform Methods
10.1 Laplace Transforms and Inverse Transforms
10.2 Transformation of Initial Value Problems
10.3 Translation and Partial Fractions
10.4 Derivatives, Integrals, and Products of Transforms
10.5 Periodic and Piecewise Continuous Input Functions
CHAPTER 11. Power Series Methods
11.1 Introduction and Review of Power Series
11.2 Power Series Solutions
11.3 Frobenius Series Solutions
11.4 Bessel Functions
References for Further Study
Appendix A: Existence and Uniqueness of Solutions
Appendix B: Theory of Determinants
Answers to Selected Problems
Index
APPLICATION MODULES
The modules listed here follow the indicated sections in the text. Most provide computing projects that illustrate the corresponding text sections. Maple, Mathematica, and MATLAB versions of these investigations are included in the Applications Manual that accompanies this textbook.
1.3 Computer-Generated Slope Fields and Solution Curves
1.4 The Logistic Equation
1.5 Indoor Temperature Oscillations
1.6 Computer Algebra Solutions
2.1 Logistic Modeling of Population Data
2.3 Rocket Propulsion
2.4 Implementing Euler's Method
2.5 Improved Euler Implementation
2.6 Runge-Kutta Implementation
3.2 Automated Row Operations
3.3 Automated Row Reduction
3.5 Automated Solution of Linear Systems
5.1 Plotting Second-Order Solution Families
5.2 Plotting Third-Order Solution Families
5.3 Approximate Solutions of Linear Equations
5.5 Automated Variation of Parameters
5.6 Forced Vibrations and Resonance
7.1 Gravitation and Kepler's Laws of Planetary Motion
7.3 Automatic Calculation of Eigenvalues and Eigenvectors
7.4 Earthquake-Induced Vibrations of Multistory Buildings
7.5 Defective Eigenvalues and Generalized Eigenvectors
7.6 Comets and Spacecraft
8.1 Automated Matrix Exponential Solutions
8.2 Automated Variation of Parameters
9.1 Phase Portraits and First-Order Equations
9.2 Phase Portraits of Almost Linear Systems
9.3 Your Own Wildlife Conservation Preserve
9.4 The Rayleigh and van der Pol Equations