(Most chapters end with a Chapter Summary, Review Problems and Group Projects.) | |

Introduction | |

Background | |

Solutions and Initial Value Problems | |

Direction Fields | |

The Approximation Method of Euler | |

First Order Differential Equations | |

Introduction: Motion of a Falling Body | |

Separable Equations | |

Linear Equations | |

Exact Equations | |

Special Integrating Factors | |

Substitutions and Transformations | |

Mathematical Models and Numerical Methods Involving First Order Equations | |

Mathematical Modeling | |

Compartmental Analysis | |

Heating and Cooling of Buildings | |

Newtonian Mechanics | |

Electrical Circuits | |

Improved Euler's Method | |

Higher-Order Numerical Methods: Taylor and Runge-Kutta | |

Linear Second Order Equations | |

Introduction: The Mass-Spring Oscillator | |

Homogeneous Linear Equations | |

The General Solution | |

Auxiliary Equations with Complex Roots | |

Nonhomogeneous Equations: the Method of Undetermined Coefficients | |

The Superposition Principle and Undetermined Coefficients Revisited | |

Variation of Parameters | |

Qualitative Considerations for Variable-Coefficient and Nonlinear Equations | |

A Closer Look at Free Mechanical Vibrations | |

A Closer Look at Forced Mechanical Vibrations | |

Introduction to Systems and Phase Plane Analysis | |

Interconnected Fluid Tanks | |

Elimination Method for Systems with Constant Coefficients | |

Solving Systems and Higher-Order Equations Numerically | |

Introduction to the Phase Plane | |

Coupled Mass-Spring Systems | |

Electrical Systems | |

Dynamical Systems, PoincarĂ© Maps, and Chaos | |

Theory of Higher-Order Linear Differential Equations | |

Basic Theory of Linear Differential Equations | |

Homogeneous Linear Equations with Constant Coefficients | |

Undetermined Coefficients and the Annihilator Method | |

Method of Variation of Parameters | |

Laplace Transforms | |

Introduction: A Mixing Problem | |

Definition of the Laplace Transform | |

Properties of the Laplace Transform | |

Inverse Laplace Transform | |

Solving Initial Value Problems | |

Transforms of Discontinuous and Periodic Functions | |

Convolution | |

Impulses and the Dirac Delta Function | |

Solving Linear Systems with Laplace Transforms | |

Series Solutions of Differential Equations | |

Introduction: The Taylor Polynomial Approximation | |

Power Series and Analytic Functions | |

Power Series Solutions to Linear Differential Equations | |

Equations with Analytic Coefficients | |

Cauchy-Euler (Equidimensional) Equations | |

Method of Frobenius | |

Finding a Second Linearly Independent Solution | |

Special Functions | |

Matrix Methods for Linear Systems | |

Introduction | |

Linear Algebraic Equations | |

Matrices and Vectors | |

Linear Systems in Normal Form | |

Homogeneous Linear Systems with Constant Coefficients | |

Complex Eigenvalues | |

Nonhomogeneous Linear Systems | |

The Matrix Exponential Function | |

Partial Differential Equations | |

Introduction: A Model for Heat Flow | |

Method of Separation of Variables | |

Fourier Series | |

Fourier Cosine and Sine Series | |

The Heat Equation | |

The Wave Equation | |

Laplace's Equation | |

Appendices | |

Newton's Method | |

Simpson's Rule | |

Cramer's Rule | |

Method of Least Squares | |

Runge-Kutta Precedure for n | |

Equations | |

Answers to Odd-Numbered Problems | |

Index | |

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