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Introduction | |

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Mathematical Models, Solutions, and Direction Fields | |

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Linear Equations: Method of Integrating Factors | |

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Numerical Approximations: Euler's Method | |

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Classification of Differential Equations | |

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First Order Differential Equations | |

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Separable Equations | |

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Modeling with First Order Equations | |

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Differences between Linear and Nonlinear Equations | |

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Autonomous Equations and Population Dynamics | |

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Exact Equations and Integrating Factors | |

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Accuracy of Numerical Methods | |

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Improved Euler and Runge-Kutta Methods | |

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Projects | |

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Harvesting a Renewable Resource | |

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Designing a Drip Dispenser for a Hydrology Experiment | |

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A Mathematical Model of a Groundwater Contaminant Source | |

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Monte-Carlo Option Pricing: Pricing Financial Options by Flipping a Coin | |

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Systems of Two First Order Equations | |

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Systems of Two Linear Algebraic Equations | |

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Systems of Two First Order Linear Differential Equations | |

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Homogeneous Linear Systems with Constant Coefficients | |

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Complex Eigenvalues | |

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Repeated Eigenvalues | |

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A Brief Introduction to Nonlinear Systems | |

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Numerical Methods for Systems of First Order Equations | |

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Projects | |

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Eigenvalue Placement Design of a Satellite Attitude Control System | |

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Estimating Rate Constants for an Open Two-Compartment Model | |

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The Ray Theory of Wave Propagation | |

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A Blood-Brain Pharmacokinetic Model | |

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Second Order Linear Equations | |

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Definitions and Examples | |

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Theory of Second Order Linear Homogeneous Equations | |

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Linear Homogeneous Equations with Constant Coefficients | |

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Mechanical and Electrical Vibrations | |

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Nonhomogeneous Equations: Method of Undetermined Coefficients | |

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Forced Vibrations, Frequency Response, and Resonance | |

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Variation of Parameters | |

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Projects | |

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A Vibration Insulation Problem | |

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Linearization of a Nonlinear Mechanical System | |

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A Spring-Mass Event Problem | |

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Uniformly Distributing Points on a Sphere | |

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Euler-Lagrange Equations | |

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The Laplace Transform | |

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Definition of the Laplace Transform | |

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Properties of the Laplace Transform | |

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The Inverse Laplace Transform | |

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Solving Differential Equations with Laplace Transforms | |

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Discontinuous Functions and Periodic Functions | |

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Differential Equations with Discontinuous Forcing Functions | |

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Impulse Functions | |

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Convolution Integrals and Their Applications | |

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Linear Systems and Feedback Control | |

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Projects | |

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An Electric Circuit Problem | |

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Effects of Pole Locations on Step Responses of Second Order Systems | |

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The Watt Governor, Feedback Control, and Stability | |

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Systems of First Order Linear Equations | |

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Definitions and Examples | |

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Basic Theory of First Order Linear Systems | |

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Homogeneous Linear Systems with Constant Coefficients | |

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Complex Eigenvalues | |

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Fundamental Matrices and the Exponential of a Matrix | |

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Nonhomogeneous Linear Systems | |

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Defective Matrices | |

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Projects | |

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A Compartment Model of Heat Flow in a Rod | |

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Earthquakes and Tall Buildings | |

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Controlling a Spring-Mass System to Equilibrium | |

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Nonlinear Differential Equations and Stability | |

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Almost Linear Systems | |

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Competing Species | |

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Predator-Prey Equations | |

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Periodic Solutions and Limit Cycles | |

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Chaos and Strange Attractors: The Lorenz Equations | |

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Projects | |

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Modeling of Epidemics | |

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Harvesting in a Competitive Environment | |

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The Rossler System | |

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[Chapters 8-10 in Boundary Value Problems version only] | |

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Series Solutions of Second Order Equations | |

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Review of Power Systems | |

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Series Solutions Near an Ordinary Point, Part I | |

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Series Solutions Near an Ordinary Point, Part II | |

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Regular Singular Points | |

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

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

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Bessel's Equation | |

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Projects | |

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Distraction Through a Circular Aperture | |

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Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator | |

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Perturbation Methods | |

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Partial Differential Equations and Fourier Series | |

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Two-Point Boundary Value Problems | |

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Fourier Series | |

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The Fourier Convergence Theorem | |

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Even and Odd Functions | |

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Separation of Variables, Heat Conduction in a Rod | |

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Other Heat Conduction Problems | |

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The Wave Equation, Vibrations of an Elastic String | |

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Laplace's Equation | |

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Projects | |

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Estimating the Diffusion Coefficient in the Heat Equation | |

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The Transmission Line Problem | |

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Solving Poisson's Equation by Finite Differences | |

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Boundary Value Problems and Sturm-Liouville Theory | |

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The Occurrence of Two-Point Boundary Value Problems | |

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Sturm-Liouville Boundary Value Problems | |

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Nonhomogeneous Boundary Value Problems | |

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Singular Sturm-Liouville Problems | |

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Further Remarks on the Method of Separation of Variables: A Bessel Series | |

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Expansion | |

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Series of Orthogonal Functions: Mean Convergence | |

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Projects | |

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Dynamic Behavior of a Hanging Cable | |

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Advection-Dispersion: A Model for Solute Transport in Saturated Porous | |

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Media | |

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Fisher's Equation for Population Growth and Dispersion | |

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Matrices and Linear Algebra | |

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Matrices | |

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Systems of Linear Algebraic Equations, Linear Independence, and Rank | |

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Determinants and Inverses | |

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The Eigenvalue Problem | |

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Complex Variables | |