Differential Equations and Linear Algebra, 4th Edition

Differential Equations and Linear Algebra, 4th Edition

by C. Henry Edwards, David E. Penney, David T. Calvis, David Calvis
Released January 2017
Publisher(s): Pearson
ISBN: 9780136739692

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Book description

For courses in Differential Equations and Linear Algebra.

The right balance between concepts, visualization, applications, and skills Differential Equations and Linear Algebra provides the conceptual development and geometric visualization of a modern differential equations and linear algebra course that is essential to science and engineering students. It balances traditional manual methods with the new, computer-based methods that illuminate qualitative phenomena – a comprehensive approach that makes accessible a wider range of more realistic applications.

The book combines core topics in elementary differential equations with concepts and methods of elementary linear algebra.

It starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout.

For the first time, MyLab™ Math is available for this text, providing online homework with immediate feedback, the complete eText, and more.

Additionally, new presentation slides created by author David Calvis are available in Beamer (LaTeX) and PDF formats.

The slides are ideal for classroom lectures and student review, and combined with Calvis’ superlative instructional videos offer a level of support not found in any other Differential Equations course.

Also available with MyLab Math

MyLab Math is the teaching and learning platform that empowers you to reach every student. By combining trusted author content with digital tools and a flexible platform, MyLab Math personalizes the learning experience and improves results for each student.

Table of contents

  1. Differential Equations & Linear Algebra
  2. Contents
  3. Application Modules
  4. Preface
    1. Principal Features of This Revision
    2. Features of This Text
    3. Supplements
  5. 1 First-Order Differential Equations
    1. 1.1 Differential Equations and Mathematical Models
      1. Differential Equations and Mathematical Models
      2. Mathematical Models
      3. Examples and Terminology
        1. Solution
        2. Solution
      4. 1.1 Problems
        1. Differential Equations as Models
    2. 1.2 Integrals as General and Particular Solutions
      1. Solution
      2. Velocity and Acceleration
        1. Solution
      3. Physical Units
      4. Vertical Motion with Gravitational Acceleration
      5. A Swimmer’s Problem
      6. 1.2 Problems
        1. Velocity Given Graphically
    3. 1.3 Slope Fields and Solution Curves
      1. Slope Fields and Graphical Solutions
        1. Solution
      2. Applications of Slope Fields
      3. Existence and Uniqueness of Solutions
      4. 1.3 Problems
      5. 1.3 Application Computer-Generated Slope Fields and Solution Curves
    4. 1.4 Separable Equations and Applications
      1. Solution
      2. Implicit, General, and Singular Solutions
        1. Solution
      3. Natural Growth and Decay
      4. The Natural Growth Equation
        1. Solution
        2. Solution
      5. Cooling and Heating
        1. Solution
      6. Torricelli’s Law
        1. Solution
      7. 1.4 Problems
        1. Torricelli’s Law
      8. 1.4 Application The Logistic Equation
    5. 1.5 Linear First-Order Equations
      1. Solution
      2. Solution
      3. A Closer Look at the Method
        1. Solution
      4. Mixture Problems
        1. Solution
        2. Solution
      5. 1.5 Problems
        1. Mixture Problems
        2. Polluted Reservoir
      6. 1.5 Application Indoor Temperature Oscillations
    6. 1.6 Substitution Methods and Exact Equations
      1. Solution
      2. Homogeneous Equations
        1. Solution
        2. Solution
      3. Bernoulli Equations
      4. Flight Trajectories
        1. Solution
      5. Exact Differential Equations
        1. Solution
      6. Reducible Second-Order Equations
        1. Solution
        2. Solution
      7. 1.6 Problems
      8. 1.6 Application Computer Algebra Solutions
    7. Chapter 1 Summary
      1. Chapter 1 Review Problems
  6. 2 Mathematical Models and Numerical Methods
    1. 2.1 Population Models
      1. Bounded Populations and the Logistic Equation
      2. Limiting Populations and Carrying Capacity
        1. Solution
      3. Historical Note
      4. More Applications of the Logistic Equation
        1. Solution
      5. Doomsday versus Extinction
        1. Solution
      6. 2.1 Problems
      7. 2.1 Application Logistic Modeling of Population Data
    2. 2.2 Equilibrium Solutions and Stability
      1. Stability of Critical Points
      2. Harvesting a Logistic Population
      3. Bifurcation and Dependence on Parameters
      4. 2.2 Problems
        1. Constant-Rate Harvesting
    3. 2.3 Acceleration–Velocity Models
      1. Resistance Proportional to Velocity
        1. Solution
      2. Resistance Proportional to Square of Velocity
      3. Variable Gravitational Acceleration
        1. Solution
      4. Escape Velocity
      5. 2.3 Problems
      6. 2.3 Application Rocket Propulsion
        1. Constant Thrust
        2. No Resistance
        3. Free Space
    4. 2.4 Numerical Approximation: Euler’s Method
      1. Solution
      2. Local and Cumulative Errors
      3. A Word of Caution
        1. Solution
      4. 2.4 Problems
      5. 2.4 Application Implementing Euler’s Method
        1. Famous Numbers Investigation
    5. 2.5 A Closer Look at the Euler Method
      1. Solution
    6. An Improvement in Euler’s Method
      1. Answer
      2. 2.5 Problems
        1. Decreasing Step Size
        2. 2.5 Application Improved Euler Implementation
          1. Famous Numbers Revisited
          2. Logistic Population Investigation
          3. Periodic Harvesting and Restocking
    7. 2.6 The Runge–Kutta Method
      1. Solution
      2. 2.6 Problems
        1. Velocity-Acceleration Problems
        2. 2.6 Application Runge–Kutta Implementation
        3. Famous Numbers Revisited, One Last Time
        4. The Skydiver’s Descent
  7. 3 Linear Systems and Matrices
    1. 3.1 Introduction to Linear Systems
      1. Two Equations in Two Unknowns
      2. Three Possibilities
      3. The Method of Elimination
      4. Three Equations in Three Unknowns
        1. Solution
        2. Solution
      5. A Differential Equations Application
      6. 3.1 Problems
    2. 3.2 Matrices and Gaussian Elimination
      1. Coefficient Matrices
      2. Elementary Row Operations
      3. Echelon Matrices
      4. Gaussian Elimination
      5. 3.2 Problems
      6. 3.2 Application Automated Row Reduction
    3. 3.3 Reduced Row-Echelon Matrices
      1. Solution
      2. The Three Possibilities
      3. Homogeneous Systems
      4. Equal Numbers of Equations and Variables
      5. 3.3 Problems
      6. 3.3 Application Automated Row Reduction
    4. 3.4 Matrix Operations
      1. Vectors
      2. Matrix Multiplication
      3. Matrix Equations
      4. Matrix Algebra
      5. 3.4 Problems
    5. 3.5 Inverses of Matrices
      1. The Inverse Matrix A-1
      2. How to Find A-1
        1. Solution
      3. Matrix Equations
        1. Solution
      4. Nonsingular Matrices
      5. 3.5 Problems
      6. 3.5 Application Automated Solution of Linear Systems
    6. 3.6 Determinants
      1. Higher-Order Determinants
      2. Row and Column Properties
      3. The Transpose of a Matrix
      4. Determinants and Invertibility
      5. Cramer’s Rule for n×n Systems
        1. Solution
      6. Inverses and the Adjoint Matrix
        1. Solution
      7. Computational Efficiency
      8. 3.6 Problems
    7. 3.7 Linear Equations and Curve Fitting
      1. Solution
      2. Modeling World Population Growth
      3. Geometric Applications
        1. Solution
        2. Solution
      4. 3.7 Problems
        1. Population Modeling
  8. 4 Vector Spaces
    1. 4.1 The Vector Space R3
      1. The Vector Space R2
        1. Solution
      2. Linear Independence in R3
      3. Basis Vectors in R3
      4. Subspaces of R3
      5. 4.1 Problems
    2. 4.2 The Vector Space Rn and Subspaces
      1. Definition of a Vector Space
      2. Subspaces
      3. 4.2 Problems
    3. 4.3 Linear Combinations and Independence of Vectors
      1. Linear Independence
      2. 4.3 Problems
    4. 4.4 Bases and Dimension for Vector Spaces
      1. Bases for Solution Spaces
        1. Solution
      2. 4.4 Problems
    5. 4.5 Row and Column Spaces
      1. Row Space and Row Rank
      2. Column Space and Column Rank
        1. Solution
      3. Rank and Dimension
      4. Nonhomogeneous Linear Systems
      5. 4.5 Problems
    6. 4.6 Orthogonal Vectors in Rn
      1. Solution
      2. Orthogonal Complements
      3. 4.6 Problems
    7. 4.7 General Vector Spaces
      1. Function Spaces
        1. Solution
        2. Solution
      2. Solution Spaces of Differential Equations
      3. 4.7 Problems
  9. 5 Higher-Order Linear Differential Equations
    1. 5.1 Introduction: Second-Order Linear Equations
      1. A Typical Application
      2. Homogeneous Second-Order Linear Equations
        1. Solution
      3. Linearly Independent Solutions
      4. General Solutions
      5. Linear Second-Order Equations with Constant Coefficients
        1. Solution
      6. 5.1 Problems
      7. 5.1 Application Plotting Second-Order Solution Families
    2. 5.2 General Solutions of Linear Equations
      1. Existence and Uniqueness of Solutions
      2. Linearly Independent Solutions
        1. Solution
        2. Solution
      3. General Solutions
      4. Nonhomogeneous Equations
        1. Solution
      5. 5.2 Problems
      6. 5.2 Application Plotting Third-Order Solution Families
    3. 5.3 Homogeneous Equations with Constant Coefficients
      1. The Characteristic Equation
      2. Distinct Real Roots
        1. Solution
      3. Polynomial Differential Operators
      4. Repeated Real Roots
        1. Solution
      5. Complex-Valued Functions and Euler’s Formula
      6. Complex Roots
        1. Solution
        2. Solution
      7. Repeated Complex Roots
        1. Solution
        2. Solution
      8. 5.3 Problems
      9. 5.3 Application Approximate Solutions of Linear Equations
    4. 5.4 Mechanical Vibrations
      1. The Simple Pendulum
      2. Free Undamped Motion
        1. Solution
      3. Free Damped Motion
        1. Solution
      4. 5.4 Problems
        1. Simple Pendulum
        2. Free Damped Motion
        3. Differential Equations and Determinism
    5. 5.5 Nonhomogeneous Equations and Undetermined Coefficients
      1. Solution
      2. Solution
      3. Solution
      4. Solution
      5. The General Approach
        1. Solution
        2. Solution
        3. Solution
      6. The Case of Duplication
        1. Solution
        2. Solution
        3. Solution
      7. Variation of Parameters
        1. Solution
      8. 5.5 Problems
      9. 5.5 Application Automated Variation of Parameters
    6. 5.6 Forced Oscillations and Resonance
      1. Undamped Forced Oscillations
        1. Solution
      2. Beats
      3. Resonance
      4. Modeling Mechanical Systems
        1. Solution
        2. Solution
      5. Damped Forced Oscillations
        1. Solution
      6. 5.6 Problems
        1. Automobile Vibrations
      7. 5.6 Application Forced Vibrations
  10. 6 Eigenvalues and Eigenvectors
    1. 6.1 Introduction to Eigenvalues
      1. The Characteristic Equation
        1. Solution
        2. Solution
      2. Eigenspaces
        1. Solution
      3. 6.1 Problems
    2. 6.2 Diagonalization of Matrices
      1. Similarity and Diagonalization
      2. 6.2 Problems
    3. 6.3 Applications Involving Powers of Matrices
      1. Solution
      2. Transition Matrices
      3. Predator-Prey Models
      4. The Cayley-Hamilton Theorem
      5. 6.3 Problems
        1. Predator-Prey
  11. 7 Linear Systems of Differential Equations
    1. 7.1 First-Order Systems and Applications
      1. Initial Applications
      2. First-Order Systems
        1. Solution
      3. Simple Two-Dimensional Systems
      4. Linear Systems
      5. 7.1 Problems
      6. 7.1 Application Gravitation and Kepler’s Laws of Planetary Motion
    2. 7.2 Matrices and Linear Systems
      1. First-Order Linear Systems
      2. Independence and General Solutions
      3. Initial Value Problems and Elementary Row Operations
        1. Solution
      4. Nonhomogeneous Solutions
      5. 7.2 Problems
    3. 7.3 The Eigenvalue Method for Linear Systems
      1. The Eigenvalue Method
      2. Distinct Real Eigenvalues
        1. Solution
      3. Compartmental Analysis
        1. Solution
      4. Complex Eigenvalues
        1. Solution
        2. Solution
      5. 7.3 Problems
        1. Cascading Brine Tanks
        2. Interconnected Brine Tanks
        3. Open Three-Tank System
        4. Closed Three-Tank System
      6. 7.3 Application Automatic Calculation of Eigenvalues and Eigenvectors
    4. 7.4 A Gallery of Solution Curves of Linear Systems
      1. Systems of Dimension n=2
      2. Real Eigenvalues
      3. Saddle Points
      4. Nodes: Sinks and Sources
      5. Zero Eigenvalues and Straight-Line Solutions
      6. Repeated Eigenvalues; Proper and Improper Nodes
      7. The Special Case of a Repeated Zero Eigenvalue
      8. Complex Conjugate Eigenvalues and Eigenvectors
      9. Pure Imaginary Eigenvalues: Centers and Elliptical Orbits
        1. Solution
      10. Complex Eigenvalues: Spiral Sinks and Sources
        1. Solution
        2. Solution
      11. A 3-Dimensional Example
      12. 7.4 Problems
      13. 7.4 Application Dynamic Phase Plane Graphics
    5. 7.5 Second-Order Systems and Mechanical Applications*
      1. Solution of Second-Order Systems
      2. Forced Oscillations and Resonance
      3. Periodic and Transient Solutions
      4. 7.5 Problems
        1. The Two-Axle Automobile
      5. 7.5 Application Earthquake-Induced Vibrations of Multistory Buildings
    6. 7.6 Multiple Eigenvalue Solutions
      1. Solution
      2. Defective Eigenvalues
      3. The Case of Multiplicity k=2
        1. Solution
      4. Generalized Eigenvectors
        1. Solution
      5. The General Case
      6. An Application
      7. The Jordan Normal Form
      8. The General Cayley-Hamilton Theorem
      9. 7.6 Problems
      10. 7.6 Application Defective Eigenvalues and Generalized Eigenvectors
    7. 7.7 Numerical Methods for Systems
      1. Euler Methods for Systems
      2. The Runge–Kutta Method and Second-Order Equations
      3. Higher-Order Systems
        1. Solution
      4. Variable Step Size Methods
      5. Earth–Moon Satellite Orbits
      6. 7.7 Problems
        1. Batted Baseball
    8. 7.7 Application Comets and Spacecraft
      1. Your Spacecraft Landing
      2. Kepler’s Law of Planetary (or Satellite) Motion
      3. Halley’s Comet
      4. Your Own Comet
  12. 8 Matrix Exponential Methods
    1. 8.1 Matrix Exponentials and Linear Systems
      1. Fundamental Matrix Solutions
        1. Solution
      2. Exponential Matrices
      3. Matrix Exponential Solutions
        1. Solution
      4. General Matrix Exponentials
        1. Solution
      5. 8.1 Problems
      6. 8.1 Application Automated Matrix Exponential Solutions
    2. 8.2 Nonhomogeneous Linear Systems
      1. Undetermined Coefficients
        1. Solution
      2. Variation of Parameters
        1. Solution
      3. 8.2 Problems
        1. Two Brine Tanks
      4. 8.2 Application Automated Variation of Parameters
    3. 8.3 Spectral Decomposition Methods
      1. The Case of Distinct Eigenvalues
      2. Second-Order Linear Systems
      3. The General Case
      4. 8.3 Problems
  13. 9 Nonlinear Systems and Phenomena
    1. 9.1 Stability and the Phase Plane
      1. Solution
      2. Phase Portraits
      3. Critical Point Behavior
      4. Stability
      5. Asymptotic Stability
      6. 9.1 Problems
      7. 9.1 Application Phase Plane Portraits and First-Order Equations
    2. 9.2 Linear and Almost Linear Systems
      1. Linearization Near a Critical Point
      2. Isolated Critical Points of Linear Systems
      3. Almost Linear Systems
        1. Solution
        2. Solution
      4. 9.2 Problems
        1. Bifurcations
      5. 9.2 Application Phase Plane Portraits of Almost Linear Systems
    3. 9.3 Ecological Models: Predators and Competitors
      1. Competing Species
      2. Interactions of Logistic Populations
      3. 9.3 Problems
        1. Predator–Prey System
        2. Competition System
        3. Competition System
        4. Logistic Prey Population
        5. Doomsday vs. Extinction
      4. 9.3 Application Your Own Wildlife Conservation Preserve
    4. 9.4 Nonlinear Mechanical Systems
      1. The Position–Velocity Phase Plane
      2. Damped Nonlinear Vibrations
      3. The Nonlinear Pendulum
      4. Period of Undamped Oscillation
      5. Damped Pendulum Oscillations
      6. 9.4 Problems
        1. Critical Points for Damped Pendulum
        2. Critical Points for Mass-Spring System
        3. Critical Points for Physical Systems
        4. Period of Oscillation
      7. 9.4 Application The Rayleigh, van der Pol, and FitzHugh-Nagumo Equations
        1. Rayleigh’s Equation
        2. Van der Pol’s Equation
        3. The FitzHugh-Nagumo Equations
  14. 10 Laplace Transform Methods
    1. 10.1 Laplace Transforms and Inverse Transforms
      1. Linearity of Transforms
      2. Inverse Transforms
      3. Piecewise Continuous Functions
        1. Solution
      4. General Properties of Transforms
      5. 10.1 Problems
      6. 10.1 Application Computer Algebra Transforms and Inverse Transforms
    2. 10.2 Transformation of Initial Value Problems
      1. Solution of Initial Value Problems
        1. Solution
        2. Solution
      2. Linear Systems
        1. Solution
      3. The Transform Perspective
      4. Additional Transform Techniques
        1. Solution
        2. Solution
        3. Solution
      5. Extension of Theorem 1
      6. 10.2 Problems
      7. 10.2 Application Transforms of Initial Value Problems
    3. 10.3 Translation and Partial Fractions
      1. Solution
      2. Solution
      3. Solution
      4. Solution
      5. Resonance and Repeated Quadratic Factors
        1. Solution
        2. Solution
      6. 10.3 Problems
      7. 10.3 Application Damping and Resonance Investigations
    4. 10.4 Derivatives, Integrals, and Products of Transforms
      1. Differentiation of Transforms
        1. Solution
        2. Solution
      2. Integration of Transforms
        1. Solution
        2. Solution
      3. * Proofs of Theorems
      4. 10.4 Problems
    5. 10.5 Periodic and Piecewise Continuous Input Functions
      1. Solution
      2. Solution
      3. Solution
      4. Transforms of Periodic Functions
        1. Solution
      5. 10.5 Problems
      6. 10.5 Application Engineering Functions
  15. 11 Power Series Methods
    1. 11.1 Introduction and Review of Power Series
      1. Power Series Operations
      2. The Power Series Method
        1. Solution
      3. Shift of Index of Summation
        1. Solution
        2. Solution
        3. Solution
      4. 11.1 Problems
    2. 11.2 Power Series Solutions
      1. Solution
      2. Solution
      3. Translated Series Solutions
        1. Solution
      4. Types of Recurrence Relation
        1. Solution
      5. The Legendre Equation
      6. 11.2 Problems
      7. 11.2 Application Automatic Computation of Series Coefficients
    3. 11.3 Frobenius Series Solutions
      1. Types of Singular Points
      2. The Method of Frobenius
        1. Solution
      3. Frobenius Series Solutions
        1. Solution
        2. Solution
      4. When r1-r2 Is an Integer
        1. Solution
      5. Summary
      6. 11.3 Problems
      7. 11.3 Application Automating the Frobenius Series Method
    4. 11.4 Bessel Functions
      1. The Case r=p > 0 > 0
      2. The Case r = -p &lt; 0 < 0
      3. The Gamma Function
      4. Bessel Functions of the First Kind
      5. Bessel Functions of the Second Kind
      6. Bessel Function Identities
      7. Applications of Bessel Functions
        1. Solution
        2. Solution
      8. 11.4 Problems
  16. References for Further Study
  17. APPENDIX A Existence and Uniqueness of Solutions
    1. A.1 Existence of Solutions
    2. A.2 Linear Systems
    3. A.3 Local Existence
    4. A.4 Uniqueness of Solutions
    5. A.5 Well-Posed Problems and Mathematical Models
      1. Problems
  18. APPENDIX B Theory of Determinants
    1. Determinants and Elementary Row Operations
    2. Determinants and Invertibility
    3. Cramer’s Rule and Inverse Matrices
    4. Inverses and the Adjoint Matrix
  19. Answers to Selected Problems
    1. Chapter 1
      1. Section 1.1
      2. Section 1.2
      3. Section 1.3
      4. Section 1.4
      5. Section 1.5
      6. Section 1.6
      7. Chapter 1 Review Problems
    2. Chapter 2
      1. Section 2.1
      2. Section 2.2
      3. Section 2.3
      4. Section 2.4
      5. Section 2.5
      6. Section 2.6
    3. Chapter 3
      1. Section 3.1
      2. Section 3.2
      3. Section 3.3
      4. Section 3.4
      5. Section 3.5
      6. Section 3.6
      7. Section 3.7
    4. Chapter 4
      1. Section 4.1
      2. Section 4.2
      3. Section 4.3
      4. Section 4.4
      5. Section 4.5
      6. Section 4.6
      7. Section 4.7
    5. Chapter 5
      1. Section 5.1
      2. Section 5.2
      3. Section 5.3
      4. Section 5.4
      5. Section 5.5
      6. Section 5.6
    6. Chapter 6
      1. Section 6.1
      2. Section 6.2
      3. Section 6.3
    7. Chapter 7
      1. Section 7.1
      2. Section 7.2
      3. Section 7.3
      4. Section 7.4
      5. Section 7.5
      6. Section 7.6
      7. Section 7.7
    8. Chapter 8
      1. Section 8.1
      2. Section 8.2
      3. Section 8.3
    9. Chapter 9
      1. Section 9.1
      2. Section 9.2
      3. Section 9.3
      4. Section 9.4
    10. Chapter 10
      1. Section 10.1
      2. Section 10.2
      3. Section 10.3
      4. Section 10.4
      5. Section 10.5
    11. Chapter 11
      1. Section 11.1
      2. Section 11.2
      3. Section 11.3
      4. Section 11.4
      5. Appendix A
  20. Index
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    9. I
    10. J
    11. K
    12. L
    13. M
    14. N
    15. O
    16. P
    17. Q
    18. R
    19. S
    20. T
    21. U
    22. V
    23. W
    24. Z

Product information

  • Title: Differential Equations and Linear Algebra, 4th Edition
  • Author(s): C. Henry Edwards, David E. Penney, David T. Calvis, David Calvis
  • Release date: January 2017
  • Publisher(s): Pearson
  • ISBN: 9780136739692