Differential Equations An Introduction to Modern Methods and Applications
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
P.1 Harvesting a Renewable Resource
P.2 Designing a Drip Dispenser for a Hydrology Experiment
P.3 A Mathematical Model of a Groundwater Contaminant Source
P.4 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
P.1 Eigenvalue-Placement Design of a Satellite Attitude
Control System
P.2 Estimating Rate Constants for an Open Two-Compartment
Model
P.3 The Ray Theory of Wave Propagation
P.4 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
P.1 A Vibration Insulation Problem
P.2 Linearization of a Nonlinear Mechanical System
P.3 A Spring-Mass Event Problem
P.4 Uniformly Distributing Points on a Sphere
P.5 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
P.1 An Electric Circuit Problem
P.2 Effects of Pole Locations on Step Responses of Second
Order Systems
P.3 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
Nondefective Matrices with Complex Eigenvalues
Fundamental Matrices and the Exponential of a Matrix
Nonhomogeneous Linear Systems
Defective Matrices
Projects
P.1 A Compartment Model of Heat Flow in a Rod
P.2 Earthquakes and Tall Buildings
P.3 Controlling a Spring-Mass System to Equilibrium
Nonlinear Differential Equations and Stability
Autonomous Systems and Stability
Almost Linear Systems
Competing Species
Predator–Prey Equations
Periodic Solutions and Limit Cycles
Chaos and Strange Attractors: The Lorenz Equations
Projects
P.1 Modeling of Epidemics
P.2 Harvesting in a Competitive Environment
P.3 The Ro¨ ssler System
Matrices and Linear Algebra
A.1 Matrices
A.2 Systems of Linear Algebraic Equations,
Linear Independence,and Rank
A.3 Determinants and Inverses
A.4 The Eigenvalue Problem
B Complex Variables
ANSWERS TO SELECTED PROBLEMS
REFERENCES
PHOTO CREDITS
INDEX
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