Fundamentals of Differential Equations

Cover
Title Page
Copyright Page
Dedicated to R. Kent Nagle
Contents
Preface
	Our Goal
	New to This Edition
	Prerequisites
	Sample Syllabi
	Retained Features
	Technology and Supplements
	Acknowledgments
	Acknowledgments for the Global Edition
Chapter 1: Introduction
	1.1. Background
	1.2. Solutions and Initial Value Problems
	1.3. Direction Fields
	1.4. The Approximation Method of Euler
	Chapter 1: Summary
	Review Problems for Chapter 1
	Technical Writing Exercises for Chapter 1
	Projects for Chapter 1
		A. Picard’s Method
		B. The Phase Line
		C. Applications to Economics
		D. Taylor Series Method
Chapter 2: First-Order Differential Equations
	2.1. Introduction: Motion of a Falling Body
	2.2. Separable Equations
	2.3. Linear Equations
	2.4. Exact Equations
	2.5. Special Integrating Factors
	2.6. Substitutions and Transformations
	Chapter 2: Summary
	Review Problems for Chapter 2
	Technical Writing Exercises for Chapter 2
	Projects for Chapter 2
		A. Oil Spill in a Canal
		B. Differential Equations in Clinical Medicine
		C. Torricelli’s Law of Fluid Flow
		D. The Snowplow Problem
		E. Two Snowplows
		F. Clairaut Equations and Singular Solutions
		G. Multiple Solutions of a First-Order Initial Value Problem
		H. Utility Functions and Risk Aversion
		I. Designing a Solar Collector
		J. Asymptotic Behavior of Solutions to Linear Equations
Chapter 3: Mathematical Models and Numerical Methods Involving First-Order Equations
	3.1. Mathematical Modeling
	3.2. Compartmental Analysis
	3.3. Heating and Cooling of Buildings
	3.4. Newtonian Mechanics
	3.5. Electrical Circuits
	3.6. Numerical Methods: A Closer Look At Euler’s Algorithm
	3.7. Higher-Order Numerical Methods: Taylor and Runge–Kutta
	Projects for Chapter 3
		A. Dynamics of HIV Infection
		B. Aquaculture
		C. Curve of Pursuit
		D. Aircraft Guidance in a Crosswind
		E. Market Equilibrium: Stability and Time Paths
		F. Stability of Numerical Methods
		G. Period Doubling and Chaos
Chapter 4: Linear Second-Order Equations
	4.1. Introduction: The Mass–Spring Oscillator
	4.2. Homogeneous Linear Equations: The General Solution
	4.3. Auxiliary Equations with Complex Roots
	4.4. Nonhomogeneous Equations: The Method of Undetermined Coefficients
	4.5. The Superposition Principle and Undetermined Coefficients Revisited
	4.6. Variation of Parameters
	4.7. Variable-Coefficient Equations
	4.8. Qualitative Considerations for Variable-Coefficient and Nonlinear Equations
	4.9. A Closer Look at Free Mechanical Vibrations
	4.10. A Closer Look at Forced Mechanical Vibrations
	Chapter 4: Summary
	Review Problems for Chapter 4
	Technical Writing Exercises for Chapter 4
	Projects for Chapter 4
		A. Nonlinear Equations Solvable by First-Order Techniques
		B. Apollo Reentry
		C. Simple Pendulum
		D. Linearization of Nonlinear Problems
		E. Convolution Method
		F. Undetermined Coefficients Using Complex Arithmetic
		G. Asymptotic Behavior of Solutions
		H. Gravity Train†
Chapter 5: Introduction to Systems and Phase Plane Analysis
	5.1. Interconnected Fluid Tanks
	5.2. Differential Operators and the Elimination Method* for Systems
	5.3. Solving Systems and Higher-Order Equations Numerically
	5.4. Introduction to the Phase Plane
	5.5. Applications to Biomathematics: Epidemic and Tumor Growth Models
	5.6. Coupled Mass–Spring Systems
	5.7. Electrical Systems
	5.8. Dynamical Systems, Poincaré Maps, and Chaos
	Chapter 5: Summary
	Review Problems for Chapter 5
	Projects for Chapter 5
		A. Designing a Landing System for Interplanetary Travel
		B. Spread of Staph Infections in Hospitals—Part I
		C. Things That Bob
		D. Hamiltonian Systems
		E. Cleaning Up the Great Lakes
		F. The 2014-2015 Ebola Epidemic
		G. Phase-Locked Loops
Chapter 6: Theory of Higher-Order Linear Differential Equations
	6.1. Basic Theory of Linear Differential Equations
	6.2. Homogeneous Linear Equations with Constant Coefficients
	6.3. Undetermined Coefficients and the Annihilator Method
	6.4. Method Of Variation of Parameters
	Chapter 6: Summary
	Review Problems for Chapter 6
	Technical Writing Exercises for Chapter 6
	Projects for Chapter 6
		A. Computer Algebra Systems and Exponential Shift
		B. Justifying the Method of Undetermined Coefficients
		C. Transverse Vibrations of a Beam
		D. Higher-Order Difference Equations
Chapter 7: Laplace Transforms
	7.1. Introduction: A Mixing Problem
	7.2. Definition of the Laplace Transform
	7.3. Properties of the Laplace Transform
	7.4. Inverse Laplace Transform
	7.5. Solving Initial Value Problems
	7.6. Transforms of Discontinuous Functions
	7.7. Transforms of Periodic and Power Functions
	7.8. Convolution
	7.9. Impulses and the Dirac Delta Function
	7.10. Solving Linear Systems with Laplace Transforms
	Chapter 7: Summary
	Review Problems for Chapter 7
	Technical Writing Exercises for Chapter 7
	Projects for Chapter 7
		A. Duhamel’s Formulas
		B. Frequency Response Modeling
		C. Determining System Parameters
Chapter 8: Series Solutions of Differential Equations
	8.1. Introduction: The Taylor Polynomial Approximation
	8.2. Power Series and Analytic Functions
	8.3. Power Series Solutions to Linear Differential Equations
	8.4. Equations with Analytic Coefficients
	8.5. Cauchy–Euler (Equidimensional) Equations
	8.6. Method of Frobenius
	8.7. Finding a Second Linearly Independent Solution
	8.8. Special Functions
	Chapter 8: Summary
	Review Problems for Chapter 8
	Technical Writing Exercises for Chapter 8
	Projects for Chapter 8
		A. Alphabetization Algorithms
		B. Spherically Symmetric Solutions to Schrödinger’s Equation for the Hydrogen Atom
		C. Airy’s Equation
		D. Buckling of a Tower
		E. Aging Spring and Bessel Functions
Chapter 9: Matrix Methods for Linear Systems
	9.1. Introduction
	9.2. Review 1: Linear Algebraic Equations
	9.3. Review 2: Matrices and Vectors
	9.4. Linear Systems in Normal Form
	9.5. Homogeneous Linear Systems with Constant Coefficients
	9.6. Complex Eigenvalues
	9.7. Nonhomogeneous Linear Systems
	9.8. The Matrix Exponential Function
	Chapter 9: Summary
	Review Problems for Chapter 9
	Technical Writing Exercises for Chapter 9
	Projects for Chapter 9
		A. Uncoupling Normal Systems
		B. Matrix Laplace Transform Method
		C. Undamped Second-Order Systems
Chapter 10: Partial Differential Equations
	10.1. Introduction: A Model for Heat Flow
	10.2. Method of Separation of Variables
	10.3. Fourier Series
	10.4. Fourier Cosine and Sine Series
	10.5. The Heat Equation
	10.6. The Wave Equation
	10.7. Laplace’s Equation
	Chapter 10: Summary
	Technical Writing Exercises for Chapter 10
	Projects for Chapter 10
		A. Steady-State Temperature Distribution in a Circular Cylinder
		B. Laplace Transform Solution of the Wave Equation
		C. Green’s Function
		D. Numerical Method for Δu = f on α Rectangle
		E. The Telegrapher’s Equation and the Cable Equation
Appendices
	Appendix A: Review of Integration Techniques
	Appendix B: Newton’s Method
	Appendix C: Simpson’s Rule
	Appendix D: Cramer’s Rule
	Appendix E: Method of Least Squares
	Appendix F: Runge–Kutta Procedure for n Equations
	Appendix G: Software for Analyzing Differential Equations
Answers to Odd-Numbered Problems
Index
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