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Table of Contents
Chapter 1: Introduction
1.1 Mathematical Models and Solutions
1.2 Qualitative Methods: Phase Lines and Direction Fields
1.3 Definitions, Classification, and Terminology
Chapter 2: First Order Differential Equations
2.1 Separable Equations
2.2 Linear Equations: Method of Integrating Factors
2.3 Modeling with First Order Equations
2.4 Differences Between Linear and Nonlinear Equations
2.5 Autonomous Equations and Population Dynamics
2.6 Exact Equations and Integrating Factors
2.7 Substitution Methods
Projects
2.P.1 Harvesting a Renewable Resource
2.P.2 A Mathematical Model of a Groundwater Contaminant Source
2.P.3 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin
Chapter 3: Systems of Two First Order Equations
3.1 Systems of Two Linear Algebraic Equations
3.2 Systems of Two First Order Linear Differential Equations
3.3 Homogeneous Linear Systems with Constant Coefficients
3.4 Complex Eigenvalues
3.5 Repeated Eigenvalues
3.6 A Brief Introduction to Nonlinear Systems
Projects
3.P.1 Estimating Rate Constants for an Open TwoCompartment Model
3.P.2 A BloodBrain Pharmacokinetic Model
Chapter 4: Second Order Linear Equations
4.1 Definitions and Examples
4.2 Theory of Second Order Linear Homogeneous Equations
4.3 Linear Homogeneous Equations with Constant Coefficients
4.4 Mechanical and Electrical Vibrations
4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
4.6 Forced Vibrations, Frequency Response, and Resonance
4.7 Variation of Parameters
Projects
4.P.1 A Vibration Insulation Problem
4.P.2 Linearization of a Nonlinear Mechanical System
4.P.3 A SpringMass Event Problem
4.P.4 EulerLagrange Equations
Chapter 5: The Laplace Transform
5.1 Definition of the Laplace Transform
5.2 Properties of the Laplace Transform
5.3 The Inverse Laplace Transform
5.4 Solving Differential Equations with Laplace Transforms
5.5 Discontinuous Functions and Periodic Functions
5.6 Differential Equations with Discontinuous Forcing Functions
5.7 Impulse Functions
5.8 Convolution Integrals and Their Applications
5.9 Linear Systems and Feedback Control
Projects
5.P.1 An Electric Circuit Problem
5.P.2 The Watt Governor, Feedback Control, and Stability
Chapter 6: Systems of First Order Linear Equations
6.1 Definitions and Examples
6.2 Basic Theory of First Order Linear Systems
6.3 Homogeneous Linear Systems with Constant Coefficients
6.4 Nondefective Matrices with Complex Eigenvalues
6.5 Fundamental Matrices and the Exponential of a Matrix
6.6 Nonhomogeneous Linear Systems
6.7 Defective Matrices
Projects
6.P.1 Earthquakes and Tall Buildings
6.P.2 Controlling a SpringMass System to Equilibrium
Chapter 7: Nonlinear Differential Equations and Stability
7.1 Autonomous Systems and Stability
7.2 Almost Linear Systems
7.3 Competing Species
7.4 PredatorPrey Equations
7.5 Periodic Solutions and Limit Cycles
7.6 Chaos and Strange Attractors: The Lorenz Equations
Projects
7.P.1 Modeling of Epidemics
7.P.2 Harvesting in a Competitive Environment
7.P.3 The Rossler System
Chapter 8: Numerical Methods
8.1 Numerical Approximations: Euler’s Method
8.2 Accuracy of Numerical Methods
8.3 Improved Euler and RungeKutta Methods
8.4 Numerical Methods for Systems of First Order Equations
Projects
8.P.1 Designing a Drip Dispenser for a Hydrology Experiment
8.P.2 Monte Carlo Option Pricing: Pricing Financial Option by Flipping a Coin
Chapter 9: Series Solutions of Second order Equations
9.1 Review of Power Series
9.2 Series Solutions Near an Ordinary Point, Part I
9.3 Series Solutions Near an Ordinary Point, Part II
9.4 Regular Singular Points
9.5 Series Solutions Near a Regular Singular Point, Part I
9.6 Series Solutions Near a Regular Singular Point, Part II
9.7 Bessel’s Equation
Projects
9.P.1 Diffraction Through a Circular Aperature
9.P.2 Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator
9.P.3 Perturbation Methods
Chapter 10: Orthogonal Functions, Fourier Series and BoundaryValue Problems
10.1 Orthogonal Families in the Space PC [a,b]
10.2 Fourier Series
10.3 Elementary TwoPoint Boundary Value Problems
10.4 General SturmLiouville Boundary Value Problems
10.5 Generalized Fourier Series and Eigenfunction Expansions
10.6 Singular Boundary Value Problems
10.7 Convergence Issues
Chapter 11: Elementary Partial Differential Equations
11.1 Terminology
11.2 Heat Conduction in a Rod—Homogeneous Case
11.3 Heat Conduction in a Rod—Nonhomogeneous Case
11.4 Wave Equation—Vibrations of an Elastic String
11.5 Wave Equation—Vibrations of a Circular Membrane
11.6 Laplace Equation
Projects
11.P.1 Estimating the Diffusion Coefficient in the Heat Equation
11.P.2 The Transmission Line Problem
11.P.3 Solving Poisson’s Equation by Finite Differences
11.P.4 Dynamic Behavior of a Hanging Cable
11.P.5 Advection Dispersion: A Model for Solute Transport in Saturated Porous Media
11.P.6 Fisher’s Equation for Population Growth and Dispersion
Appendices
11.A Derivation of the Heat Equation
11.B Derivation of the Wave Equation
A: Matrices and Linear Algebra
A.1 Matrices
A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank
A.3 Determinates and Inverses
A.4 The Eigenvalue Problem
B: Complex Variables
Answers to Selected Problems
References
Index