Preface | |

Introduction 1 | |

Some Basic Mathematical Models | |

Direction Fields | |

Solutions of Some Differential Equations | |

Classification of Differential Equations | |

Historical Remarks | |

First Order Differential Equations | |

Linear Equations | |

Method of Integrating Factors | |

Separable Equations | |

Modeling with First Order Equations | |

Differences Between Linear and Nonlinear Equations | |

Autonomous Equations and Population Dynamics | |

Exact Equations and Integrating Factors | |

Numerical Approximations: Euler's Method | |

The Existence and Uniqueness Theorem | |

First Order Difference Equations | |

Second Order Linear Equations 135 | |

Homogeneous Equations with Constant Coef?cients | |

Fundamental Solutions of Linear Homogeneous Equations | |

The Wronskian | |

Complex Roots of the Characteristic Equation | |

Repeated Roots | |

Reduction of Order | |

Nonhomogeneous Equations | |

Method of Undetermined Coefficients | |

Variation of Parameters | |

Mechanical and Electrical Vibrations | |

Forced Vibrations | |

Higher Order Linear Equations | |

General Theory of nth Order Linear Equations | |

Homogeneous Equations with Constant Coef?cients | |

The Method of Undetermined Coef?cients | |

The Method of Variation of Parameters | |

Series Solutions of Second Order Linear Equations | |

Review of Power Series | |

Series Solutions Near an Ordinary Point, Part I | |

Series Solutions Near an Ordinary Point, Part II | |

Euler Equations | |

Regular Singular Points | |

Series Solutions Near a Regular Singular Point, Part I | |

Series Solutions Near a Regular Singular Point, Part II | |

Bessel's Equation | |

The Laplace Transform | |

Definition of the Laplace Transform | |

Solution of Initial Value Problems | |

Step Functions | |

Differential Equations with Discontinuous Forcing Functions | |

Impulse Functions | |

The Convolution Integral | |

Systems of First Order Linear Equations | |

Introduction | |

Review of Matrices | |

Systems of Linear Algebraic Equations | |

Linear Independence, Eigenvalues, Eigenvectors | |

Basic Theory of Systems of First Order Linear Equations | |

Homogeneous Linear Systems with Constant Coefficients | |

Complex Eigenvalues | |

Fundamental Matrices | |

Repeated Eigenvalues | |

Nonhomogeneous Linear Systems | |

Numerical Methods | |

The Euler or Tangent Line Method | |

Improvements on the Euler Method | |

The Runge-Kutta Method | |

Multistep Methods | |

More on Errors | |

Stability | |

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