Differential Equations and Linear Algebra 4th Edition by C. Henry Edwards, ISBN-13: 978-0134497181

Description

Differential Equations and Linear Algebra 4th Edition by C. Henry Edwards, ISBN-13: 978-0134497181

[PDF eBook eTextbook]

• Publisher: ‎ Pearson; 4th edition (January 4, 2017)
• Language: ‎ English
• 768 pages
• ISBN-10: ‎ 013449718X
• ISBN-13: ‎ 978-0134497181

For courses in Differential Equations and Linear Algebra .

Table of Contents:

Differential Equations & Linear Algebra

Contents

Application Modules

Preface

Principal Features of This Revision

Features of This Text

Supplements

1 First-Order Differential Equations

1.1 Differential Equations and Mathematical Models

Differential Equations and Mathematical Models

Mathematical Models

Examples and Terminology

Solution

Solution

1.1 Problems

Differential Equations as Models

1.2 Integrals as General and Particular Solutions

Solution

Velocity and Acceleration

Solution

Physical Units

Vertical Motion with Gravitational Acceleration

A Swimmer’s Problem

1.2 Problems

Velocity Given Graphically

1.3 Slope Fields and Solution Curves

Slope Fields and Graphical Solutions

Solution

Applications of Slope Fields

Existence and Uniqueness of Solutions

1.3 Problems

1.3 Application Computer-Generated Slope Fields and Solution Curves

1.4 Separable Equations and Applications

Solution

Implicit, General, and Singular Solutions

Solution

Natural Growth and Decay

The Natural Growth Equation

Solution

Solution

Cooling and Heating

Solution

Torricelli’s Law

Solution

1.4 Problems

Torricelli’s Law

1.4 Application The Logistic Equation

1.5 Linear First-Order Equations

Solution

Solution

A Closer Look at the Method

Solution

Mixture Problems

Solution

Solution

1.5 Problems

Mixture Problems

Polluted Reservoir

1.5 Application Indoor Temperature Oscillations

1.6 Substitution Methods and Exact Equations

Solution

Homogeneous Equations

Solution

Solution

Bernoulli Equations

Flight Trajectories

Solution

Exact Differential Equations

Solution

Reducible Second-Order Equations

Solution

Solution

1.6 Problems

1.6 Application Computer Algebra Solutions

Chapter 1 Summary

Chapter 1 Review Problems

2 Mathematical Models and Numerical Methods

2.1 Population Models

Bounded Populations and the Logistic Equation

Limiting Populations and Carrying Capacity

Solution

Historical Note

More Applications of the Logistic Equation

Solution

Doomsday versus Extinction

Solution

2.1 Problems

2.1 Application Logistic Modeling of Population Data

2.2 Equilibrium Solutions and Stability

Stability of Critical Points

Harvesting a Logistic Population

Bifurcation and Dependence on Parameters

2.2 Problems

Constant-Rate Harvesting

2.3 Acceleration–Velocity Models

Resistance Proportional to Velocity

Solution

Resistance Proportional to Square of Velocity

Variable Gravitational Acceleration

Solution

Escape Velocity

2.3 Problems

2.3 Application Rocket Propulsion

Constant Thrust

No Resistance

Free Space

2.4 Numerical Approximation: Euler’s Method

Solution

Local and Cumulative Errors

A Word of Caution

Solution

2.4 Problems

2.4 Application Implementing Euler’s Method

Famous Numbers Investigation

2.5 A Closer Look at the Euler Method

Solution

An Improvement in Euler’s Method

Answer

2.5 Problems

Decreasing Step Size

2.5 Application Improved Euler Implementation

Famous Numbers Revisited

Logistic Population Investigation

Periodic Harvesting and Restocking

2.6 The Runge–Kutta Method

Solution

2.6 Problems

Velocity-Acceleration Problems

2.6 Application Runge–Kutta Implementation

Famous Numbers Revisited, One Last Time

The Skydiver’s Descent

3 Linear Systems and Matrices

3.1 Introduction to Linear Systems

Two Equations in Two Unknowns

Three Possibilities

The Method of Elimination

Three Equations in Three Unknowns

Solution

Solution

A Differential Equations Application

3.1 Problems

3.2 Matrices and Gaussian Elimination

Coefficient Matrices

Elementary Row Operations

Echelon Matrices

Gaussian Elimination

3.2 Problems

3.2 Application Automated Row Reduction

3.3 Reduced Row-Echelon Matrices

Solution

The Three Possibilities

Homogeneous Systems

Equal Numbers of Equations and Variables

3.3 Problems

3.3 Application Automated Row Reduction

3.4 Matrix Operations

Vectors

Matrix Multiplication

Matrix Equations

Matrix Algebra

3.4 Problems

3.5 Inverses of Matrices

The Inverse Matrix A-1

How to Find A-1

Solution

Matrix Equations

Solution

Nonsingular Matrices

3.5 Problems

3.5 Application Automated Solution of Linear Systems

3.6 Determinants

Higher-Order Determinants

Row and Column Properties

The Transpose of a Matrix

Determinants and Invertibility

Cramer’s Rule for n×n Systems

Solution

Inverses and the Adjoint Matrix

Solution

Computational Efficiency

3.6 Problems

3.7 Linear Equations and Curve Fitting

Solution

Modeling World Population Growth

Geometric Applications

Solution

Solution

3.7 Problems

Population Modeling

4 Vector Spaces

4.1 The Vector Space R3

The Vector Space R2

Solution

Linear Independence in R3

Basis Vectors in R3

Subspaces of R3

4.1 Problems

4.2 The Vector Space Rn and Subspaces

Definition of a Vector Space

Subspaces

4.2 Problems

4.3 Linear Combinations and Independence of Vectors

Linear Independence

4.3 Problems

4.4 Bases and Dimension for Vector Spaces

Bases for Solution Spaces

Solution

4.4 Problems

4.5 Row and Column Spaces

Row Space and Row Rank

Column Space and Column Rank

Solution

Rank and Dimension

Nonhomogeneous Linear Systems

4.5 Problems

4.6 Orthogonal Vectors in Rn

Solution

Orthogonal Complements

4.6 Problems

4.7 General Vector Spaces

Function Spaces

Solution

Solution

Solution Spaces of Differential Equations

4.7 Problems

5 Higher-Order Linear Differential Equations

5.1 Introduction: Second-Order Linear Equations

A Typical Application

Homogeneous Second-Order Linear Equations

Solution

Linearly Independent Solutions

General Solutions

Linear Second-Order Equations with Constant Coefficients

Solution

5.1 Problems

5.1 Application Plotting Second-Order Solution Families

5.2 General Solutions of Linear Equations

Existence and Uniqueness of Solutions

Linearly Independent Solutions

Solution

Solution

General Solutions

Nonhomogeneous Equations

Solution

5.2 Problems

5.2 Application Plotting Third-Order Solution Families

5.3 Homogeneous Equations with Constant Coefficients

The Characteristic Equation

Distinct Real Roots

Solution

Polynomial Differential Operators

Repeated Real Roots

Solution

Complex-Valued Functions and Euler’s Formula

Complex Roots

Solution

Solution

Repeated Complex Roots

Solution

Solution

5.3 Problems

5.3 Application Approximate Solutions of Linear Equations

5.4 Mechanical Vibrations

The Simple Pendulum

Free Undamped Motion

Solution

Free Damped Motion

Solution

5.4 Problems

Simple Pendulum

Free Damped Motion

Differential Equations and Determinism

5.5 Nonhomogeneous Equations and Undetermined Coefficients

Solution

Solution

Solution

Solution

The General Approach

Solution

Solution

Solution

The Case of Duplication

Solution

Solution

Solution

Variation of Parameters

Solution

5.5 Problems

5.5 Application Automated Variation of Parameters

5.6 Forced Oscillations and Resonance

Undamped Forced Oscillations

Solution

Beats

Resonance

Modeling Mechanical Systems

Solution

Solution

Damped Forced Oscillations

Solution

5.6 Problems

Automobile Vibrations

5.6 Application Forced Vibrations

6 Eigenvalues and Eigenvectors

6.1 Introduction to Eigenvalues

The Characteristic Equation

Solution

Solution

Eigenspaces

Solution

6.1 Problems

6.2 Diagonalization of Matrices

Similarity and Diagonalization

6.2 Problems

6.3 Applications Involving Powers of Matrices

Solution

Transition Matrices

Predator-Prey Models

The Cayley-Hamilton Theorem

6.3 Problems

Predator-Prey

7 Linear Systems of Differential Equations

7.1 First-Order Systems and Applications

Initial Applications

First-Order Systems

Solution

Simple Two-Dimensional Systems

Linear Systems

7.1 Problems

7.1 Application Gravitation and Kepler’s Laws of Planetary Motion

7.2 Matrices and Linear Systems

First-Order Linear Systems

Independence and General Solutions

Initial Value Problems and Elementary Row Operations

Solution

Nonhomogeneous Solutions

7.2 Problems

7.3 The Eigenvalue Method for Linear Systems

The Eigenvalue Method

Distinct Real Eigenvalues

Solution

Compartmental Analysis

Solution

Complex Eigenvalues

Solution

Solution

7.3 Problems

Cascading Brine Tanks

Interconnected Brine Tanks

Open Three-Tank System

Closed Three-Tank System

7.3 Application Automatic Calculation of Eigenvalues and Eigenvectors

7.4 A Gallery of Solution Curves of Linear Systems

Systems of Dimension n=2

Real Eigenvalues

Saddle Points

Nodes: Sinks and Sources

Zero Eigenvalues and Straight-Line Solutions

Repeated Eigenvalues; Proper and Improper Nodes

The Special Case of a Repeated Zero Eigenvalue

Complex Conjugate Eigenvalues and Eigenvectors

Pure Imaginary Eigenvalues: Centers and Elliptical Orbits

Solution

Complex Eigenvalues: Spiral Sinks and Sources

Solution

Solution

A 3-Dimensional Example

7.4 Problems

7.4 Application Dynamic Phase Plane Graphics

7.5 Second-Order Systems and Mechanical Applications*

Solution of Second-Order Systems

Forced Oscillations and Resonance

Periodic and Transient Solutions

7.5 Problems

The Two-Axle Automobile

7.5 Application Earthquake-Induced Vibrations of Multistory Buildings

7.6 Multiple Eigenvalue Solutions

Solution

Defective Eigenvalues

The Case of Multiplicity k=2

Solution

Generalized Eigenvectors

Solution

The General Case

An Application

The Jordan Normal Form

The General Cayley-Hamilton Theorem

7.6 Problems

7.6 Application Defective Eigenvalues and Generalized Eigenvectors

7.7 Numerical Methods for Systems

Euler Methods for Systems

The Runge–Kutta Method and Second-Order Equations

Higher-Order Systems

Solution

Variable Step Size Methods

Earth–Moon Satellite Orbits

7.7 Problems

Batted Baseball

7.7 Application Comets and Spacecraft

Your Spacecraft Landing

Kepler’s Law of Planetary (or Satellite) Motion

Halley’s Comet

Your Own Comet

8 Matrix Exponential Methods

8.1 Matrix Exponentials and Linear Systems

Fundamental Matrix Solutions

Solution

Exponential Matrices

Matrix Exponential Solutions

Solution

General Matrix Exponentials

Solution

8.1 Problems

8.1 Application Automated Matrix Exponential Solutions

8.2 Nonhomogeneous Linear Systems

Undetermined Coefficients

Solution

Variation of Parameters

Solution

8.2 Problems

Two Brine Tanks

8.2 Application Automated Variation of Parameters

8.3 Spectral Decomposition Methods

The Case of Distinct Eigenvalues

Second-Order Linear Systems

The General Case

8.3 Problems

9 Nonlinear Systems and Phenomena

9.1 Stability and the Phase Plane

Solution

Phase Portraits

Critical Point Behavior

Stability

Asymptotic Stability

9.1 Problems

9.1 Application Phase Plane Portraits and First-Order Equations

9.2 Linear and Almost Linear Systems

Linearization Near a Critical Point

Isolated Critical Points of Linear Systems

Almost Linear Systems

Solution

Solution

9.2 Problems

Bifurcations

9.2 Application Phase Plane Portraits of Almost Linear Systems

9.3 Ecological Models: Predators and Competitors

Competing Species

Interactions of Logistic Populations

9.3 Problems

Predator–Prey System

Competition System

Competition System

Logistic Prey Population

Doomsday vs. Extinction

9.3 Application Your Own Wildlife Conservation Preserve

9.4 Nonlinear Mechanical Systems

The Position–Velocity Phase Plane

Damped Nonlinear Vibrations

The Nonlinear Pendulum

Period of Undamped Oscillation

Damped Pendulum Oscillations

9.4 Problems

Critical Points for Damped Pendulum

Critical Points for Mass-Spring System

Critical Points for Physical Systems

Period of Oscillation

9.4 Application The Rayleigh, van der Pol, and FitzHugh-Nagumo Equations

Rayleigh’s Equation

Van der Pol’s Equation

The FitzHugh-Nagumo Equations

10 Laplace Transform Methods

10.1 Laplace Transforms and Inverse Transforms

Linearity of Transforms

Inverse Transforms

Piecewise Continuous Functions

Solution

General Properties of Transforms

10.1 Problems

10.1 Application Computer Algebra Transforms and Inverse Transforms

10.2 Transformation of Initial Value Problems

Solution of Initial Value Problems

Solution

Solution

Linear Systems

Solution

The Transform Perspective

Additional Transform Techniques

Solution

Solution

Solution

Extension of Theorem 1

10.2 Problems

10.2 Application Transforms of Initial Value Problems

10.3 Translation and Partial Fractions

Solution

Solution

Solution

Solution

Resonance and Repeated Quadratic Factors

Solution

Solution

10.3 Problems

10.3 Application Damping and Resonance Investigations

10.4 Derivatives, Integrals, and Products of Transforms

Differentiation of Transforms

Solution

Solution

Integration of Transforms

Solution

Solution

* Proofs of Theorems

10.4 Problems

10.5 Periodic and Piecewise Continuous Input Functions

Solution

Solution

Solution

Transforms of Periodic Functions

Solution

10.5 Problems

10.5 Application Engineering Functions

11 Power Series Methods

11.1 Introduction and Review of Power Series

Power Series Operations

The Power Series Method

Solution

Shift of Index of Summation

Solution

Solution

Solution

11.1 Problems

11.2 Power Series Solutions

Solution

Solution

Translated Series Solutions

Solution

Types of Recurrence Relation

Solution

The Legendre Equation

11.2 Problems

11.2 Application Automatic Computation of Series Coefficients

11.3 Frobenius Series Solutions

Types of Singular Points

The Method of Frobenius

Solution

Frobenius Series Solutions

Solution

Solution

When r1-r2 Is an Integer

Solution

Summary

11.3 Problems

11.3 Application Automating the Frobenius Series Method

11.4 Bessel Functions

The Case r=p > 0

The Case r = -p < 0

The Gamma Function

Bessel Functions of the First Kind

Bessel Functions of the Second Kind

Bessel Function Identities

Applications of Bessel Functions

Solution

Solution

11.4 Problems

References for Further Study

APPENDIX A Existence and Uniqueness of Solutions

A.1 Existence of Solutions

A.2 Linear Systems

A.3 Local Existence

A.4 Uniqueness of Solutions

A.5 Well-Posed Problems and Mathematical Models

Problems

APPENDIX B Theory of Determinants

Determinants and Elementary Row Operations

Determinants and Invertibility

Cramer’s Rule and Inverse Matrices

Inverses and the Adjoint Matrix

Answers to Selected Problems

C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia’s honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution’s highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.

David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran’s Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee’s research team’s primary focus was on the active transport of sodium ions by biological membranes. Penney’s primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He was the author of research papers in number theory and topology and was the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.

David T. Calvis is Professor of Mathematics at Baldwin Wallace University near Cleveland, Ohio. He completed a Ph.D. in complex analysis from the University of Michigan in 1988 under the direction of Fred Gehring. While at Michigan he also received a Master’s degree in Computer, Information, and Control Engineering. Having initially served at Hillsdale College in Michigan, he has been at Baldwin Wallace since 1990, most recently assisting with the creation of an Applied Mathematics program there. He has received a number of teaching awards, including BWU’s Strosacker Award for Excellence in Teaching and Student Senate Teaching Award. He is the author of a number of materials dealing with the use of computer algebra systems in mathematics instruction, and has extensive classroom experience teaching differential equations and related topics.

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