First-Order Differential Equations | |

Differential Equations and Mathematical Models | |

Integrals as General and Particular Solutions | |

Slope Fields and Solution Curves | |

Separable Equations and Applications | |

Linear First-Order Equations | |

Substitution Methods and Exact Equations | |

Mathematical Models and Numerical Methods | |

Population Models | |

Equilibrium Solutions and Stability | |

Accelerationndash;Velocity Models | |

Numerical Approximation: Euler's Method | |

A Closer Look at the Euler Method | |

The Rungendash;Kutta Method | |

Linear Systems and Matrices | |

Introduction to Linear Systems | |

Matrices and Gaussian Elimination | |

Reduced Row-Echelon Matrices | |

Matrix Operations | |

Inverses of Matrices | |

Determinants | |

Linear Equations and Curve Fitting | |

Vector Spaces | |

The Vector Space R3 | |

The Vector Space Rn and Subspaces | |

Linear Combinations and Independence of Vectors | |

Bases and Dimension for Vector Spaces | |

Row and Column Spaces | |

Orthogonal Vectors in Rn | |

General Vector Spaces | |

Higher-Order Linear Differential Equations | |

Introduction: Second-Order Linear Equations | |

General Solutions of Linear Equations | |

Homogeneous Equations with Constant Coefficients | |

Mechanical Vibrations | |

Nonhomogeneous Equations and Undetermined Coefficients | |

Forced Oscillations and Resonance | |

Eigenvalues and Eigenvectors | |

Introduction to Eigenvalues | |

Diagonalization of Matrices | |

Applications Involving Powers of Matrices | |

Linear Systems of Differential Equations | |

First-Order Systems and Applications | |

Matrices and Linear Systems | |

The Eigenvalue Method for Linear Systems | |

Second-Order Systems and Mechanical Applications | |

Multiple Eigenvalue Solutions | |

Numerical Methods for Systems | |

Matrix Exponential Methods | |

Matrix Exponentials and Linear Systems | |

Nonhomogeneous Linear Systems | |

Spectral Decomposition Methods | |

Nonlinear Systems and Phenomena | |

Stability and the Phase Plane | |

Linear and Almost Linear Systems | |

Ecological Models: Predators and Competitors | |

Nonlinear Mechanical Systems | |

Laplace Transform Methods | |

Laplace Transforms and Inverse Transforms | |

Transformation of Initial Value Problems | |

Translation and Partial Fractions | |

Derivatives, Integrals, and Products of Transforms | |

Periodic and Piecewise Continuous Input Functions | |

Power Series Methods | |

Introduction and Review of Power Series | |

Power Series Solutions | |

Frobenius Series Solutions | |

| |

Existence and Uniqueness of Solutions | |

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 | |

Computer-Generated Slope Fields and Solution Curves | |

The Logistic Equation | |

Indoor Temperature Oscillations | |

Computer Algebra Solutions | |

Logistic Modeling of Population Data | |

Rocket Propulsion | |

Implementing Euler's Method | |

Improved Euler Implementation | |

Runge-Kutta Implementation | |

Automated Row Operations | |

Automated Row Reduction | |

Automated Solution of Linear Systems | |

Plotting Seco | |

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