| |
| |
| |
| Introduction | |
| |
| |
| |
| Mathematical Models, Solutions, and Direction Fields | |
| |
| |
| |
| Linear Equations: Method of Integrating Factors | |
| |
| |
| |
| Numerical Approximations: Euler's Method | |
| |
| |
| |
| Classification of Differential Equations | |
| |
| |
| |
| First Order Differential Equations | |
| |
| |
| |
| Separable Equations | |
| |
| |
| |
| Modeling with First Order Equations | |
| |
| |
| |
| Differences between Linear and Nonlinear Equations | |
| |
| |
| |
| Autonomous Equations and Population Dynamics | |
| |
| |
| |
| Exact Equations and Integrating Factors | |
| |
| |
| |
| Accuracy of Numerical Methods | |
| |
| |
| |
| Improved Euler and Runge-Kutta Methods | |
| |
| |
| Projects | |
| |
| |
| |
| Harvesting a Renewable Resource | |
| |
| |
| |
| Designing a Drip Dispenser for a Hydrology Experiment | |
| |
| |
| |
| A Mathematical Model of a Groundwater Contaminant Source | |
| |
| |
| |
| Monte-Carlo Option Pricing: Pricing Financial Options by Flipping a Coin | |
| |
| |
| |
| Systems of Two First Order Equations | |
| |
| |
| |
| Systems of Two Linear Algebraic Equations | |
| |
| |
| |
| Systems of Two First Order Linear Differential Equations | |
| |
| |
| |
| Homogeneous Linear Systems with Constant Coefficients | |
| |
| |
| |
| Complex Eigenvalues | |
| |
| |
| |
| Repeated Eigenvalues | |
| |
| |
| |
| A Brief Introduction to Nonlinear Systems | |
| |
| |
| |
| Numerical Methods for Systems of First Order Equations | |
| |
| |
| Projects | |
| |
| |
| |
| Eigenvalue Placement Design of a Satellite Attitude Control System | |
| |
| |
| |
| Estimating Rate Constants for an Open Two-Compartment Model | |
| |
| |
| |
| The Ray Theory of Wave Propagation | |
| |
| |
| |
| A Blood-Brain Pharmacokinetic Model | |
| |
| |
| |
| Second Order Linear Equations | |
| |
| |
| |
| Definitions and Examples | |
| |
| |
| |
| Theory of Second Order Linear Homogeneous Equations | |
| |
| |
| |
| Linear Homogeneous Equations with Constant Coefficients | |
| |
| |
| |
| Mechanical and Electrical Vibrations | |
| |
| |
| |
| Nonhomogeneous Equations: Method of Undetermined Coefficients | |
| |
| |
| |
| Forced Vibrations, Frequency Response, and Resonance | |
| |
| |
| |
| Variation of Parameters | |
| |
| |
| Projects | |
| |
| |
| |
| A Vibration Insulation Problem | |
| |
| |
| |
| Linearization of a Nonlinear Mechanical System | |
| |
| |
| |
| A Spring-Mass Event Problem | |
| |
| |
| |
| Uniformly Distributing Points on a Sphere | |
| |
| |
| |
| Euler-Lagrange Equations | |
| |
| |
| |
| The Laplace Transform | |
| |
| |
| |
| Definition of the Laplace Transform | |
| |
| |
| |
| Properties of the Laplace Transform | |
| |
| |
| |
| The Inverse Laplace Transform | |
| |
| |
| |
| Solving Differential Equations with Laplace Transforms | |
| |
| |
| |
| Discontinuous Functions and Periodic Functions | |
| |
| |
| |
| Differential Equations with Discontinuous Forcing Functions | |
| |
| |
| |
| Impulse Functions | |
| |
| |
| |
| Convolution Integrals and Their Applications | |
| |
| |
| |
| Linear Systems and Feedback Control | |
| |
| |
| Projects | |
| |
| |
| |
| An Electric Circuit Problem | |
| |
| |
| |
| Effects of Pole Locations on Step Responses of Second Order Systems | |
| |
| |
| |
| The Watt Governor, Feedback Control, and Stability | |
| |
| |
| |
| Systems of First Order Linear Equations | |
| |
| |
| |
| Definitions and Examples | |
| |
| |
| |
| Basic Theory of First Order Linear Systems | |
| |
| |
| |
| Homogeneous Linear Systems with Constant Coefficients | |
| |
| |
| |
| Complex Eigenvalues | |
| |
| |
| |
| Fundamental Matrices and the Exponential of a Matrix | |
| |
| |
| |
| Nonhomogeneous Linear Systems | |
| |
| |
| |
| Defective Matrices | |
| |
| |
| Projects | |
| |
| |
| |
| A Compartment Model of Heat Flow in a Rod | |
| |
| |
| |
| Earthquakes and Tall Buildings | |
| |
| |
| |
| Controlling a Spring-Mass System to Equilibrium | |
| |
| |
| |
| Nonlinear Differential Equations and Stability | |
| |
| |
| |
| Almost Linear Systems | |
| |
| |
| |
| Competing Species | |
| |
| |
| |
| Predator-Prey Equations | |
| |
| |
| |
| Periodic Solutions and Limit Cycles | |
| |
| |
| |
| Chaos and Strange Attractors: The Lorenz Equations | |
| |
| |
| Projects | |
| |
| |
| |
| Modeling of Epidemics | |
| |
| |
| |
| Harvesting in a Competitive Environment | |
| |
| |
| |
| The Rossler System | |
| |
| |
| [Chapters 8-10 in Boundary Value Problems version only] | |
| |
| |
| |
| Series Solutions of Second Order Equations | |
| |
| |
| |
| Review of Power Systems | |
| |
| |
| |
| Series Solutions Near an Ordinary Point, Part I | |
| |
| |
| |
| Series Solutions Near an Ordinary Point, Part II | |
| |
| |
| |
| Regular Singular Points | |
| |
| |
| |
| Series Solutions Near a Regular Singular Point, Part I | |
| |
| |
| |
| Series Solutions Near a Regular Singular Point, Part II | |
| |
| |
| |
| Bessel's Equation | |
| |
| |
| Projects | |
| |
| |
| |
| Distraction Through a Circular Aperture | |
| |
| |
| |
| Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator | |
| |
| |
| |
| Perturbation Methods | |
| |
| |
| |
| Partial Differential Equations and Fourier Series | |
| |
| |
| |
| Two-Point Boundary Value Problems | |
| |
| |
| |
| Fourier Series | |
| |
| |
| |
| The Fourier Convergence Theorem | |
| |
| |
| |
| Even and Odd Functions | |
| |
| |
| |
| Separation of Variables, Heat Conduction in a Rod | |
| |
| |
| |
| Other Heat Conduction Problems | |
| |
| |
| |
| The Wave Equation, Vibrations of an Elastic String | |
| |
| |
| |
| Laplace's Equation | |
| |
| |
| Projects | |
| |
| |
| |
| Estimating the Diffusion Coefficient in the Heat Equation | |
| |
| |
| |
| The Transmission Line Problem | |
| |
| |
| |
| Solving Poisson's Equation by Finite Differences | |
| |
| |
| |
| Boundary Value Problems and Sturm-Liouville Theory | |
| |
| |
| |
| The Occurrence of Two-Point Boundary Value Problems | |
| |
| |
| |
| Sturm-Liouville Boundary Value Problems | |
| |
| |
| |
| Nonhomogeneous Boundary Value Problems | |
| |
| |
| |
| Singular Sturm-Liouville Problems | |
| |
| |
| |
| Further Remarks on the Method of Separation of Variables: A Bessel Series | |
| |
| |
| Expansion | |
| |
| |
| |
| Series of Orthogonal Functions: Mean Convergence | |
| |
| |
| Projects | |
| |
| |
| |
| Dynamic Behavior of a Hanging Cable | |
| |
| |
| |
| Advection-Dispersion: A Model for Solute Transport in Saturated Porous | |
| |
| |
| Media | |
| |
| |
| |
| Fisher's Equation for Population Growth and Dispersion | |
| |
| |
| |
| Matrices and Linear Algebra | |
| |
| |
| |
| Matrices | |
| |
| |
| |
| Systems of Linear Algebraic Equations, Linear Independence, and Rank | |
| |
| |
| |
| Determinants and Inverses | |
| |
| |
| |
| The Eigenvalue Problem | |
| |
| |
| |
| Complex Variables | |