Differential Equations and Linear Algebra

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
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
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
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
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
6. Eigenvalues and Eigenvectors
   • 6.1 Introduction to Eigenvalues
   • 6.2 Diagonalization of Matrices
   • 6.3 Applications Involving Powers of Matrices
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 A Gallery of Solution Curves of Linear Systems
   • 7.5 Second-Order Systems and Mechanical Applications
   • 7.6 Multiple Eigenvalue Solutions
   • 7.7 Numerical Methods for Systems
8. Matrix Exponential Methods
   • 8.1 Matrix Exponentials and Linear Systems
   • 8.2 Nonhomogeneous Linear Systems
   • 8.3 Spectral Decomposition Methods
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
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
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

    Appendices

    • A: Existence and Uniqueness of Solutions
    • B: Theory of Determinants

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APPLICATION MODULES

The modules listed below follow the indicated sections in the text. Most provide computing projects that illustrate the corresponding text sections. Many of these modules are enhanced by the supplementary material found at the new Expanded Applications website.

• 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 Dynamic Phase Plane Graphics
• 7.5 Earthquake-Induced Vibrations of Multistory Buildings
• 7.6 Defective Eigenvalues and Generalized Eigenvectors
• 7.7 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