| |
| |
| Preface | |
| |
| |
| |
| Introduction to Differential Equations | |
| |
| |
| |
| Definitions and Concepts | |
| |
| |
| |
| Solutions of Differential Equations | |
| |
| |
| |
| Initial and Boundary-Value Problems | |
| |
| |
| |
| Direction Fields | |
| |
| |
| |
| First-Order Ordinary Differential Equations | |
| |
| |
| |
| Theory of First-Order Equations: A Brief Discussion | |
| |
| |
| |
| Separation of Variables | |
| |
| |
| Application: Kidney Dialysis | |
| |
| |
| |
| Homogeneous Equations | |
| |
| |
| Application: Models of Pursuit | |
| |
| |
| |
| Exact Equations | |
| |
| |
| |
| Linear Equations | |
| |
| |
| |
| Integrating Factor Approach | |
| |
| |
| |
| Variation of Parameters and the Method of Undetermined Coefficients | |
| |
| |
| Application: Antibiotic Production | |
| |
| |
| |
| Numerical Approximations of Solutions to First-Order Equations | |
| |
| |
| |
| Built-In Methods | |
| |
| |
| Application: Modeling the Spread of a Disease | |
| |
| |
| |
| Other Numerical Methods | |
| |
| |
| |
| Applications of First-Order Ordinary Differential Equations | |
| |
| |
| |
| Orthogonal Trajectories | |
| |
| |
| Application: Oblique Trajectories | |
| |
| |
| |
| Population Growth and Decay | |
| |
| |
| |
| The Malthus Model | |
| |
| |
| |
| The Logistic Equation | |
| |
| |
| Application: Harvesting | |
| |
| |
| Application: The Logistic Difference Equation | |
| |
| |
| |
| Newton's Law of Cooling | |
| |
| |
| |
| Free-Falling Bodies | |
| |
| |
| |
| Higher-Order Differential Equations | |
| |
| |
| |
| Preliminary Definitions and Notation | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| The nth-Order Ordinary Linear Differential Equation | |
| |
| |
| |
| Fundamental Set of Solutions | |
| |
| |
| |
| Existence of a Fundamental Set of Solutions | |
| |
| |
| |
| Reduction of Order | |
| |
| |
| |
| Solving Homogeneous Equations with Constant Coefficients | |
| |
| |
| |
| Second-Order Equations | |
| |
| |
| |
| Higher-Order Equations | |
| |
| |
| Application: Testing for Diabetes | |
| |
| |
| |
| Introduction to Solving Nonhomogeneous Equations with Constant Coefficients | |
| |
| |
| |
| Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients | |
| |
| |
| |
| Second-Order Equations | |
| |
| |
| |
| Higher-Order Equations | |
| |
| |
| |
| Nonhomogeneous Equations with Constant Coefficients: Variation of Parameters | |
| |
| |
| |
| Second-Order Equations | |
| |
| |
| |
| Higher-Order Nonhomogeneous Equations | |
| |
| |
| |
| Cauchy-Euler Equations | |
| |
| |
| |
| Second-Order Cauchy-Euler Equations | |
| |
| |
| |
| Higher-Order Cauchy-Euler Equations | |
| |
| |
| |
| Variation of Parameters | |
| |
| |
| |
| Series Solutions | |
| |
| |
| |
| Power Series Solutions about Ordinary Points | |
| |
| |
| |
| Series Solutions about Regular Singular Points | |
| |
| |
| |
| Method of Frobenius | |
| |
| |
| Application: Zeros of the Bessel Functions of the First Kind | |
| |
| |
| Application: The Wave Equation on a Circular Plate | |
| |
| |
| |
| Nonlinear Equations | |
| |
| |
| |
| Applications of Higher-Order Differential Equations | |
| |
| |
| |
| Harmonic Motion | |
| |
| |
| |
| Simple Harmonic Motion | |
| |
| |
| |
| Damped Motion | |
| |
| |
| |
| Forced Motion | |
| |
| |
| |
| Soft Springs | |
| |
| |
| |
| Hard Springs | |
| |
| |
| |
| Aging Springs | |
| |
| |
| Application: Hearing Beats and Resonance | |
| |
| |
| |
| The Pendulum Problem | |
| |
| |
| |
| Other Applications | |
| |
| |
| |
| L-R-C Circuits | |
| |
| |
| |
| Deflection of a Beam | |
| |
| |
| |
| Bode Plots | |
| |
| |
| |
| The Catenary | |
| |
| |
| |
| Systems of Ordinary Differential Equations | |
| |
| |
| |
| Review of Matrix Algebra and Calculus | |
| |
| |
| |
| Defining Nested Lists, Matrices, and Vectors | |
| |
| |
| |
| Extracting Elements of Matrices | |
| |
| |
| |
| Basic Computations with Matrices | |
| |
| |
| |
| Eigenvalues and Eigenvectors | |
| |
| |
| |
| Matrix Calculus | |
| |
| |
| |
| Systems of Equations: Preliminary Definitions and Theory | |
| |
| |
| |
| Preliminary Theory | |
| |
| |
| |
| Linear Systems | |
| |
| |
| |
| Homogeneous Linear Systems with Constant Coefficients | |
| |
| |
| |
| Distinct Real Eigenvalues | |
| |
| |
| |
| Complex Conjugate Eigenvalues | |
| |
| |
| |
| Alternate Method for Solving Initial-Value Problems | |
| |
| |
| |
| Repeated Eigenvalues | |
| |
| |
| |
| Nonhomogeneous First-Order Systems: Undetermined Coefficients, Variation of Parameters, and the Matrix Exponential | |
| |
| |
| |
| Undetermined Coefficients | |
| |
| |
| |
| Variation of Parameters | |
| |
| |
| |
| The Matrix Exponential | |
| |
| |
| |
| Numerical Methods | |
| |
| |
| |
| Built-In Methods | |
| |
| |
| Application: Controlling the Spread of a Disease | |
| |
| |
| |
| Euler's Method | |
| |
| |
| |
| Runge-Kutta Method | |
| |
| |
| |
| Nonlinear Systems, Linearization, and Classification of Equilibrium Points | |
| |
| |
| |
| Real Distinct Eigenvalues | |
| |
| |
| |
| Repeated Eigenvalues | |
| |
| |
| |
| Complex Conjugate Eigenvalues | |
| |
| |
| |
| Nonlinear Systems | |
| |
| |
| |
| Applications of Systems of Ordinary Differential Equations | |
| |
| |
| |
| Mechanical and Electrical Problems with First-Order Linear Systems | |
| |
| |
| |
| L-R-C Circuits with Loops | |
| |
| |
| |
| L-R-C Circuit with One Loop | |
| |
| |
| |
| L-R-C Circuit with Two Loops | |
| |
| |
| |
| Spring-Mass Systems | |
| |
| |
| |
| Diffusion and Population Problems with First-Order Linear Systems | |
| |
| |
| |
| Diffusion through a Membrane | |
| |
| |
| |
| Diffusion through a Double-Walled Membrane | |
| |
| |
| |
| Population Problems | |
| |
| |
| |
| Applications that Lead to Nonlinear Systems | |
| |
| |
| |
| Biological Systems: Predator-Prey Interactions, The Lotka-Volterra System, and Food Chains in the Chemostat | |
| |
| |
| |
| Physical Systems: Variable Damping | |
| |
| |
| |
| Differential Geometry: Curvature | |
| |
| |
| |
| Laplace Transform Methods | |
| |
| |
| |
| The Laplace Transform | |
| |
| |
| |
| Definition of the Laplace Transform | |
| |
| |
| |
| Exponential Order | |
| |
| |
| |
| Properties of the Laplace Transform | |
| |
| |
| |
| The Inverse Laplace Transform | |
| |
| |
| |
| Definition of the Inverse Laplace Transform | |
| |
| |
| |
| Laplace Transform of an Integral | |
| |
| |
| |
| Solving Initial-Value Problems with the Laplace Transform | |
| |
| |
| |
| Laplace Transforms of Step and Periodic Functions | |
| |
| |
| |
| Piecewise-Defined Functions: The Unit Step Function | |
| |
| |
| |
| Solving Initial-Value Problems | |
| |
| |
| |
| Periodic Functions | |
| |
| |
| |
| Impulse Functions: The Delta Function | |
| |
| |
| |
| The Convolution Theorem | |
| |
| |
| |
| The Convolution Theorem | |
| |
| |
| |
| Integral and Integrodifferential Equations | |
| |
| |
| |
| Applications of Laplace Transforms, Part I | |
| |
| |
| |
| Spring-Mass Systems Revisited | |
| |
| |
| |
| L-R-C Circuits Revisited | |
| |
| |
| |
| Population Problems Revisited | |
| |
| |
| Application: The Tautochrone | |
| |
| |
| |
| Laplace Transform Methods for Systems | |
| |
| |
| |
| Applications of Laplace Transforms, Part II | |
| |
| |
| |
| Coupled Spring-Mass Systems | |
| |
| |
| |
| The Double Pendulum | |
| |
| |
| Application: Free Vibration of a Three-Story Building | |
| |
| |
| |
| Eigenvalue Problems and Fourier Series | |
| |
| |
| |
| Boundary-Value Problems, Eigenvalue Problems, Sturm-Liouville Problems | |
| |
| |
| |
| Boundary-Value Problems | |
| |
| |
| |
| Eigenvalue Problems | |
| |
| |
| |
| Sturm-Liouville Problems | |
| |
| |
| |
| Fourier Sine Series and Cosine Series | |
| |
| |
| |
| Fourier Sine Series | |
| |
| |
| |
| Fourier Cosine Series | |
| |
| |
| |
| Fourier Series | |
| |
| |
| |
| Fourier Series | |
| |
| |
| |
| Even, Odd, and Periodic Extensions | |
| |
| |
| |
| Differentiation and Integration of Fourier Series | |
| |
| |
| |
| Parseval's Equality | |
| |
| |
| |
| Generalized Fourier Series | |
| |
| |
| |
| Partial Differential Equations | |
| |
| |
| |
| Introduction to Partial Differential Equations and Separation of Variables | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Separation of Variables | |
| |
| |
| |
| The One-Dimensional Heat Equation | |
| |
| |
| |
| The Heat Equation with Homogeneous Boundary Conditions | |
| |
| |
| |
| Nonhomogeneous Boundary Conditions | |
| |
| |
| |
| Insulated Boundary | |
| |
| |
| |
| The One-Dimensional Wave Equation | |
| |
| |
| |
| The Wave Equation | |
| |
| |
| |
| D'Alembert's Solution | |
| |
| |
| |
| Problems in Two Dimensions: Laplace's Equation | |
| |
| |
| |
| Laplace's Equation | |
| |
| |
| |
| Two-Dimensional Problems in a Circular Region | |
| |
| |
| |
| Laplace's Equation in a Circular Region | |
| |
| |
| |
| The Wave Equation in a Circular Region | |
| |
| |
| |
| Other Partial Differential Equations | |
| |
| |
| |
| Getting Started | |
| |
| |
| Introduction to Mathematica | |
| |
| |
| A Note Regarding Different Versions of Mathematica | |
| |
| |
| Getting Started with Mathematica | |
| |
| |
| Five Basic Rules of Mathematica Syntax | |
| |
| |
| Loading Packages | |
| |
| |
| A Word of Caution | |
| |
| |
| Getting Help from Mathematica | |
| |
| |
| Mathematica Help | |
| |
| |
| The Mathematica Menu | |
| |
| |
| Bibliography | |
| |
| |
| Index | |