Elementary Differential Equations and Boundary Value Problems

Cover
Introduction
Copyright
Dedication
About the Authors
Preface
Table of Contents
	Ch 1 Introduction
		1.1 Some Basic Mathematical Models; Direction Fields
			1.1 Problems
		1.2 Solutions of Some Differential Equations
			1.2 Problems
		1.3 Classification of Differential Equations
			1.3 Problems
		1.4 Historical Remarks
	Ch 2 First Order Differential Equations
		2.1 Linear Equations with Variable Coefficients
			2.1 Problems
		2.2 Separable Equations
			2.2 Problems
		2.3 Modeling with First Order Equations
			2.3 Problems
		2.4 Differences Between Linear and Nonlinear Equations
			2.4 Problems
		2.5 Autonomous Equations and Population Dynamics
			2.5 Problems
		2.6 Exact Equations and Integrating Factors
			2.6 Problems
		2.7 Numerical Approximations: Euler’s Method
			2.7 Problems
		2.8 The Existence and Uniqueness Theorem
			2.8 Problems
		2.9 First Order Difference Equations
			2.9 Problems
	Ch 3 Second Order Linear Equations
		3.1 Homogeneous Equations with Constant Coefficients
			3.1 Problems
		3.2 Fundamental Solutions of Linear Homogeneous Equations
			3.2 Problems
		3.3 Linear Independence and the Wronskian
			3.3 Problems
		3.4 Complex Roots of the Characteristic Equation
			3.4 Problems
		3.5 Repeated Roots; Reduction of Order
			3.5 Problems
		3.6 Nonhomogeneous Equations; Method of Undetermined Coefficients
			3.6 Problems
		3.7 Variation of Parameters
			3.7 Problems
		3.8 Mechanical and Electrical Vibrations
			3.8 Problems
		3.9 Forced Vibrations
			3.9 Problems
	Ch 4 Higher Order Linear Equations
		4.1 General Theory of nth Order Linear Equations
			4.1 Problems
		4.2 Homogeneous Equations with Constant Coeffients
			4.2 Problems
		4.3 The Method of Undetermined Coefficients
			4.3 Problems
		4.4 The Method of Variation of Parameters
			4.4 Problems
	Ch 5 Series Solutions of Second Order Linear Equations
		5.1 Review of Power Series
			5.1 Problems
		5.2 Series Solutions near an Ordinary Point, Part I
			5.2 Problems
		5.3 Series Solutions near an Ordinary Point, Part II
			5.3 Problems
		5.4 Regular Singular Points
			5.4 Problems
		5.5 Euler Equations
			5.5 Problems
		5.6 Series Solutions near a Regular Singular Point, Part I
			5.6 Problems
		5.7 Series Solutions near a Regular Singular Point, Part II
			5.7 Problems
		5.8 Bessel’s Equation
			5.8 Problems
	Ch 6 The Laplace Transform
		6.1 Definition of the Laplace Transform
			6.1 Problems
		6.2 Solution of Initial Value Problems
			6.2 Problems
		6.3 Step Functions
			6.3 Problems
		6.4 Differential Equations with Discontinuous Forcing Functions
			6.4 Problems
		6.5 Impulse Functions
			6.5 Problems
		6.6 The Convolution Integral
			6.6 Problems
	Ch 7 Systems of First Order Linear Equations
		7.1 Introduction
			7.1 Problems
		7.2 Review of Matrices
			7.2 Problems
		7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
			7.3 Problems
		7.4 Basic Theory of Systems of First Order Linear Equations
			7.4 Problems
		7.5 Homogeneous Linear Systems with Constant Coefficients
			7.5 Problems
		7.6 Complex Eigenvalues
			7.6 Problems
		7.7 Fundamental Matrices
			7.7 Problems
		7.8 Repeated Eigenvalues
			7.8 Problems
		7.9 Nonhomogeneous Linear Systems
			7.9 Problems
	Ch 8 Numerical Methods
		8.1 The Euler or Tangent Line Method
			8.1 Problems
		8.2 Improvements on the Euler Method
			8.2 Problems
		8.3 The Runge–Kutta Method
			8.3 Problems
		8.4 Multistep Methods
			8.4 Problems
		8.5 More on Errors; Stability
			8.5 Problems
		8.6 Systems of First Order Equations
			8.6 Problems
	Ch 9 Nonlinear Differential Equations and Stability
		9.1 The Phase Plane; Linear Systems
			9.1 Problems
		9.2 Autonomous Systems and Stability
			9.2 Problems
		9.3 Almost Linear Systems
			9.3 Problems
		9.4 Competing Species
			9.4 Problems
		9.5 Predator–Prey Equations
			9.5 Problems
		9.6 Liapunov’s Second Method
			9.6 Problems
		9.7 Periodic Solutions and Limit Cycles
			9.7 Problems
		9.8 Chaos and Strange Attractors; the Lorenz Equations
			9.8 Problems
	Ch 10 Partial Differential Equations and Fourier Series
		10.1 Two-Point Boundary Valve Problems
			10.1 Problems
		10.2 Fourier Series
			10.2 Problems
		10.3 The Fourier Convergence Theorem
			10.3 Problems
		10.4 Even and Odd Functions
			10.4 Problems
		10.5 Separation of Variables; Heat Conduction in a Rod
			10.5 Problems
		10.6 Other Heat Conduction Problems
			10.6 Problems
		10.7 The Wave Equation; Vibrations of an Elastic String
			10.7 Problems
		10.8 Laplace’s Equation
			10.8 Problems
			Appendix A. Derivation of the Heat Conduction Equation
			Appendix B. Derivation of the Wave Equation
	Ch 11 Boundary Value Problems and Sturm–Liouville Theory
		11.1 The Occurrence of Two Point Boundary Value Problems
			11.1 Problems
		11.2 Sturm–Liouville Boundary Value Problems
			11.2 Problems
		11.3 Nonhomogeneous Boundary Value Problems
			11.3 Problems
		11.4 Singular Sturm–Liouville Problems
			11.4 Problems
		11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
			11.5 Problems
		11.6 Series of Orthogonal Functions: Mean Convergence
			11.6 Problems
Answers
	Ch 1
	Ch 2
	Ch 3
	Ch 4
	Ch 5
	Ch 6
	Ch 7
	Ch 8
	Ch 9
	Ch 10
	Ch 11
Index
SSM - Introduction
	SSM - Table of Contents
		Ch 1
		Ch 2
		Ch 3
		Ch 4
		Ch 5
		Ch 6
		Ch 7
		Ch 8
		Ch 9
		Ch 10
		Ch 11
Student Solutions Manual
ODE Workbook Introduction
	copyright
	Workbook Preface
	Acknowledgments
	Modules/Chapters Overview
	Contents
	1. Modeling with the ODE Architect
		Building a Model of the Pacific Sardine Population
		The Logistic Equation
		Introducing Harvesting via Landing Data
		How to Model in Eight Steps
		Explorations
	2. Introduction to ODEs
		Differential Equations
		Solutions to Differential Equations
		Solving a Differential Equation
		Slope Fields
		Initial Values
		Finding a Solution Formula
		Modeling
		The Juggler
		The Sky Diver
		Explorations
	3. Some Cool ODEs
		Newton\'s Law of Cooling
		Cooling an Egg
		Finding a General Solution
		Time-Dependent Outside Temperature
		Air Conditioning a Room
		The Case of the Melting Snowman
		Explorations
	4. Second-Order Linear Equations
		Second-Order ODEs and the Architect
		Undamped Oscillations
		The Effect of Damping
		Forced Oscillations
		Beats
		Electrical Oscillations: An Analogy
		Seismographs
		Explorations
	5. Models of Motion
		Vectors
		Forces and Newton\'s Laws
		Dunk Tank
		Longer to Rise or to Fall?
		Indiana Newton
		Ski Jumping
		Explorations
	6. First-Order Linear Systems
		Background
		Examples of Systems: Pizza and Video, Coupled Springs
		Linear Systems with Constant Coefficients
		Solution Formulas: Eigenvalues and Eigenvectors
		Calculating Eigenvalues and Eigenvectors
		Phase Portraits
		Using ODE Architect to Find Eigenvalues and Eigenvectors
		Separatrices
		Parameter Movies
		Explorations
	7. Nonlinear Systems
		Linear vs. Nonlinear
		The Geometry of Nonlinear Systems
		Linearization
		Separatrices and Saddle Points
		Behavior of Solutions Away from Equilibrium Points
		Bifurcation to a Limit Cycle
		Higher Dimensions
		Spinning Bodies: Stability of Steady Rotations
		The Planar Double Pendulum
		Explorations
	8. Compartment Models
		Lake Polution
		Allergy Relief
		Lead in the Body
		Equilibrium
		The Autocatalator and a Hopf Bifurcation
		Explorations
	9. Population Models
		Modeling Population Growth
		The Logistic Model
		Two-Species Population Models
		Predator and Prey
		Species Competition
		Mathematical Epidemiology: The SIR Model
		Explorations
	10. The Pendulum and Its Friends
		Modeling Pendulum Motion
		Conservative Systems: Integrals of Motion
		The Effect of Damping
		Separatrices
		Pumping a Swing
		Writing the Equations of Motion for Pumping a Swing
		Geodesics
		Geodesics on a Surface of Revolution
		Geodesics on a Torus
		Explorations
	11. Applications of Series Solutions
		Infinite Series
		Recurrence Formulas
		Ordinary Points
		Regular Singular Points
		Bessel Functions
		Transforming Bessel\'s Equation to the Aging Spring Equation
		Explorations
	12. Chaos and Control
		Introduction
		Solutions as Functions of Time
		Poincare Sections
		Periodic Points
		The Unforced Pendulum
		The Damped Forced Pendlum
		Tangled Basins, the Wada Property
		Gaining Control
		Explorations
	13. Discrete Dynamical Systems
		Equilibrium States
		Linear vs. Nonlinear Dynamics
		Stability of a Discrete Dynamical System
		Bifurcations
		Periodic and Chaotic Dynamics
		What is Chaos?
		Complex Numbers and Functions
		Iterating a Complex Function
		Julia Sets, the Mandelbrot Set, and Cantor Dust
		Explorations
	Glossary