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Differential Equations: From Calculus to Dynamical Systems: Second Edition
 
Virginia W. Noonburg University of Hartford, West Hartford, CT
Differential Equations: From Calculus to Dynamical Systems
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6329-8
Product Code:  TEXT/43.S
List Price: $75.00
MAA Member Price: $56.25
AMS Member Price: $56.25
eBook ISBN:  978-1-4704-5108-0
Product Code:  TEXT/43.E
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
Softcover ISBN:  978-1-4704-6329-8
eBook: ISBN:  978-1-4704-5108-0
Product Code:  TEXT/43.S.B
List Price: $144.00 $109.50
MAA Member Price: $108.00 $82.13
AMS Member Price: $108.00 $82.13
Differential Equations: From Calculus to Dynamical Systems
Click above image for expanded view
Differential Equations: From Calculus to Dynamical Systems: Second Edition
Virginia W. Noonburg University of Hartford, West Hartford, CT
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6329-8
Product Code:  TEXT/43.S
List Price: $75.00
MAA Member Price: $56.25
AMS Member Price: $56.25
eBook ISBN:  978-1-4704-5108-0
Product Code:  TEXT/43.E
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
Softcover ISBN:  978-1-4704-6329-8
eBook ISBN:  978-1-4704-5108-0
Product Code:  TEXT/43.S.B
List Price: $144.00 $109.50
MAA Member Price: $108.00 $82.13
AMS Member Price: $108.00 $82.13
  • Book Details
     
     
    AMS/MAA Textbooks
    Volume: 432019; 402 pp
    MSC: Primary 34; 35

    This is a Revised Edition of: TEXT/25

    A thoroughly modern textbook for the sophomore-level differential equations course. The examples and exercises emphasize modeling not only in engineering and physics but also in applied mathematics and biology. There is an early introduction to numerical methods and, throughout, a strong emphasis on the qualitative viewpoint of dynamical systems. Bifurcations and analysis of parameter variation is a persistent theme.

    Presuming previous exposure to only two semesters of calculus, necessary linear algebra is developed as needed. The exposition is very clear and inviting. The book would serve well for use in a flipped-classroom pedagogical approach or for self-study for an advanced undergraduate or beginning graduate student.

    This second edition of Noonburg's best-selling textbook includes two new chapters on partial differential equations, making the book usable for a two-semester sequence in differential equations. It includes exercises, examples, and extensive student projects taken from the current mathematical and scientific literature.

    Ancillaries:

    Readership

    Undergraduate students interested in teaching and learning differential equations (both ordinary and PDE).

  • Table of Contents
     
     
    • Title page
    • Copyright
    • Contents
    • Preface
    • Acknowledgments
    • Chapter 1. Introduction to Differential Equations
    • 1.1. Basic Terminology
    • 1.1.1. Ordinary vs. Partial Differential Equations
    • 1.1.2. Independent Variables, Dependent Variables, and Parameters
    • 1.1.3. Order of a Differential Equation
    • 1.1.4. What is a Solution?
    • 1.1.5. Systems of Differential Equations
    • 1.2. Families of Solutions, Initial-Value Problems
    • 1.3. Modeling with Differential Equations
    • Chapter 2. First-order Differential Equations
    • 2.1. Separable First-order Equations
    • 2.1.1. Application 1: Population Growth
    • 2.1.2. Application 2: Newton’s Law of Cooling
    • 2.2. Graphical Methods, the Slope Field
    • 2.2.1. Using Graphical Methods to Visualize Solutions
    • 2.3. Linear First-order Differential Equations
    • 2.3.1. Application: Single-compartment Mixing Problem
    • 2.4. Existence and Uniqueness of Solutions
    • 2.5. More Analytic Methods for Nonlinear First-order Equations
    • 2.5.1. Exact Differential Equations
    • 2.5.2. Bernoulli Equations
    • 2.5.3. Using Symmetries of the Slope Field
    • 2.6. Numerical Methods
    • 2.6.1. Euler’s Method
    • 2.6.2. Improved Euler Method
    • 2.6.3. Fourth-order Runge-Kutta Method
    • 2.7. Autonomous Equations, the Phase Line
    • 2.7.1. Stability—Sinks, Sources, and Nodes
    • Bifurcation in Equations with Parameters
    • Chapter 3. Second-order Differential Equations
    • 3.1. General Theory of Homogeneous Linear Equations
    • 3.2. Homogeneous Linear Equations with Constant Coefficients
    • 3.2.1. Second-order Equation with Constant Coefficients
    • 3.2.2. Equations of Order Greater Than Two
    • 3.3. The Spring-mass Equation
    • 3.3.1. Derivation of the Spring-mass Equation
    • 3.3.2. The Unforced Spring-mass System
    • 3.4. Nonhomogeneous Linear Equations
    • 3.4.1. Method of Undetermined Coefficients
    • 3.4.2. Variation of Parameters
    • 3.5. The Forced Spring-mass System
    • Beats and Resonance
    • 3.6. Linear Second-order Equations with Nonconstant Coefficients
    • 3.6.1. The Cauchy-Euler Equation
    • 3.6.2. Series Solutions
    • 3.7. Autonomous Second-order Differential Equations
    • 3.7.1. Numerical Methods
    • 3.7.2. Autonomous Equations and the Phase Plane
    • Chapter 4. Linear Systems ofFirst-order Differential Equations
    • 4.1. Introduction to Systems
    • 4.1.1. Writing Differential Equations as a First-order System
    • 4.1.2. Linear Systems
    • 4.2. Matrix Algebra
    • 4.3. Eigenvalues and Eigenvectors
    • 4.4. Analytic Solutions of the Linear System ⃗𝐱’=𝐀⃗𝐱
    • 4.4.1. Application 1: Mixing Problem with Two Compartments
    • 4.4.2. Application 2: Double Spring-mass System
    • 4.5. Large Linear Systems; the Matrix Exponential
    • 4.5.1. Definition and Properties of the Matrix Exponential
    • 4.5.2. Using the Matrix Exponential to Solve a Nonhomogeneous System
    • 4.5.3. Application: Mixing Problem with Three Compartments
    • Chapter 5. Geometry of Autonomous Systems
    • 5.1. The Phase Plane for Autonomous Systems
    • 5.2. Geometric Behavior of Linear Autonomous Systems
    • 5.2.1. Linear Systems with Real (Distinct, Nonzero) Eigenvalues
    • 5.2.2. Linear Systems with Complex Eigenvalues
    • 5.2.3. The Trace-determinant Plane
    • 5.2.4. The Special Cases
    • 5.3. Geometric Behavior of Nonlinear Autonomous Systems
    • 5.3.1. Finding the Equilibrium Points
    • 5.3.2. Determining the Type of an Equilibrium
    • 5.3.3. A Limit Cycle—the Van der Pol Equation
    • 5.4. Bifurcations for Systems
    • 5.4.1. Bifurcation in a Spring-mass Model
    • 5.4.2. Bifurcation of a Predator-prey Model
    • 5.4.3. Bifurcation Analysis Applied to a Competing Species Model
    • 5.5. Student Projects
    • 5.5.1. The Wilson-Cowan Equations
    • 5.5.2. A New Predator-prey Equation—Putting It All Together
    • Chapter 6. Laplace Transforms
    • 6.1. Definition and Some Simple Laplace Transforms
    • 6.1.1. Four Simple Laplace Transforms
    • 6.1.2. Linearity of the Laplace Transform
    • 6.1.3. Transforming the Derivative of 𝑓(𝑡)
    • 6.2. Solving Equations, the Inverse Laplace Transform
    • 6.2.1. Partial Fraction Expansions
    • 6.3. Extending the Table
    • 6.3.1. Inverting a Term with an Irreducible Quadratic Denominator
    • 6.3.2. Solving Linear Systems with Laplace Transforms
    • 6.4. The Unit Step Function
    • 6.5. Convolution and the Impulse Function
    • 6.5.1. The Convolution Integral
    • 6.5.2. The Impulse Function
    • 6.5.3. Impulse Response of a Linear, Time-invariant System
    • Chapter 7. Introduction to Partial Differential Equations
    • 7.1. Solving Partial Differential Equations
    • 7.1.1. An Overview of the Method of Separation of Variables
    • 7.2. Orthogonal Functions and Trigonometric Fourier Series
    • 7.2.1. Orthogonal Families of Functions
    • 7.2.2. Properties of Fourier Series, Cosine and Sine Series
    • 7.3. Boundary-Value Problems: Sturm-Liouville otoc { } Equations
    • Chapter 8. Solving Second-order Partial Differential Equations
    • 8.1. Classification of Linear Second-order Partial Differential Equations
    • 8.2. The 1-dimensional Heat Equation
    • 8.2.1. Solution of the Heat Equation by Separation of Variables
    • 8.2.2. Other Boundary Conditions for the Heat Equation
    • 8.3. The 1-dimensional Wave Equation
    • 8.3.1. Solution of the Wave Equation by Separation of Variables
    • 8.3.2. D’Alembert’s Solution of the Wave Equation on an Infinite Interval
    • 8.4. Numerical Solution of Parabolic and Hyperbolic Equations
    • 8.5. Laplace’s Equation
    • 8.6. Student Project: Harvested Diffusive Logistic Equation
    • Appendix
    • Appendix A. Answers to Odd-numbered Exercises
    • Appendix B. Derivative and Integral Formulas
    • Appendix C. Cofactor Method for Determinants
    • Appendix D. Cramer’s Rule for Solving Systems of Linear Equations
    • Appendix E. The Wronskian
    • Appendix F. Table of Laplace Transforms
    • Appendix G. Review of Partial Derivatives
    • Index
    • Back Cover
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 432019; 402 pp
MSC: Primary 34; 35

This is a Revised Edition of: TEXT/25

A thoroughly modern textbook for the sophomore-level differential equations course. The examples and exercises emphasize modeling not only in engineering and physics but also in applied mathematics and biology. There is an early introduction to numerical methods and, throughout, a strong emphasis on the qualitative viewpoint of dynamical systems. Bifurcations and analysis of parameter variation is a persistent theme.

Presuming previous exposure to only two semesters of calculus, necessary linear algebra is developed as needed. The exposition is very clear and inviting. The book would serve well for use in a flipped-classroom pedagogical approach or for self-study for an advanced undergraduate or beginning graduate student.

This second edition of Noonburg's best-selling textbook includes two new chapters on partial differential equations, making the book usable for a two-semester sequence in differential equations. It includes exercises, examples, and extensive student projects taken from the current mathematical and scientific literature.

Ancillaries:

Readership

Undergraduate students interested in teaching and learning differential equations (both ordinary and PDE).

  • Title page
  • Copyright
  • Contents
  • Preface
  • Acknowledgments
  • Chapter 1. Introduction to Differential Equations
  • 1.1. Basic Terminology
  • 1.1.1. Ordinary vs. Partial Differential Equations
  • 1.1.2. Independent Variables, Dependent Variables, and Parameters
  • 1.1.3. Order of a Differential Equation
  • 1.1.4. What is a Solution?
  • 1.1.5. Systems of Differential Equations
  • 1.2. Families of Solutions, Initial-Value Problems
  • 1.3. Modeling with Differential Equations
  • Chapter 2. First-order Differential Equations
  • 2.1. Separable First-order Equations
  • 2.1.1. Application 1: Population Growth
  • 2.1.2. Application 2: Newton’s Law of Cooling
  • 2.2. Graphical Methods, the Slope Field
  • 2.2.1. Using Graphical Methods to Visualize Solutions
  • 2.3. Linear First-order Differential Equations
  • 2.3.1. Application: Single-compartment Mixing Problem
  • 2.4. Existence and Uniqueness of Solutions
  • 2.5. More Analytic Methods for Nonlinear First-order Equations
  • 2.5.1. Exact Differential Equations
  • 2.5.2. Bernoulli Equations
  • 2.5.3. Using Symmetries of the Slope Field
  • 2.6. Numerical Methods
  • 2.6.1. Euler’s Method
  • 2.6.2. Improved Euler Method
  • 2.6.3. Fourth-order Runge-Kutta Method
  • 2.7. Autonomous Equations, the Phase Line
  • 2.7.1. Stability—Sinks, Sources, and Nodes
  • Bifurcation in Equations with Parameters
  • Chapter 3. Second-order Differential Equations
  • 3.1. General Theory of Homogeneous Linear Equations
  • 3.2. Homogeneous Linear Equations with Constant Coefficients
  • 3.2.1. Second-order Equation with Constant Coefficients
  • 3.2.2. Equations of Order Greater Than Two
  • 3.3. The Spring-mass Equation
  • 3.3.1. Derivation of the Spring-mass Equation
  • 3.3.2. The Unforced Spring-mass System
  • 3.4. Nonhomogeneous Linear Equations
  • 3.4.1. Method of Undetermined Coefficients
  • 3.4.2. Variation of Parameters
  • 3.5. The Forced Spring-mass System
  • Beats and Resonance
  • 3.6. Linear Second-order Equations with Nonconstant Coefficients
  • 3.6.1. The Cauchy-Euler Equation
  • 3.6.2. Series Solutions
  • 3.7. Autonomous Second-order Differential Equations
  • 3.7.1. Numerical Methods
  • 3.7.2. Autonomous Equations and the Phase Plane
  • Chapter 4. Linear Systems ofFirst-order Differential Equations
  • 4.1. Introduction to Systems
  • 4.1.1. Writing Differential Equations as a First-order System
  • 4.1.2. Linear Systems
  • 4.2. Matrix Algebra
  • 4.3. Eigenvalues and Eigenvectors
  • 4.4. Analytic Solutions of the Linear System ⃗𝐱’=𝐀⃗𝐱
  • 4.4.1. Application 1: Mixing Problem with Two Compartments
  • 4.4.2. Application 2: Double Spring-mass System
  • 4.5. Large Linear Systems; the Matrix Exponential
  • 4.5.1. Definition and Properties of the Matrix Exponential
  • 4.5.2. Using the Matrix Exponential to Solve a Nonhomogeneous System
  • 4.5.3. Application: Mixing Problem with Three Compartments
  • Chapter 5. Geometry of Autonomous Systems
  • 5.1. The Phase Plane for Autonomous Systems
  • 5.2. Geometric Behavior of Linear Autonomous Systems
  • 5.2.1. Linear Systems with Real (Distinct, Nonzero) Eigenvalues
  • 5.2.2. Linear Systems with Complex Eigenvalues
  • 5.2.3. The Trace-determinant Plane
  • 5.2.4. The Special Cases
  • 5.3. Geometric Behavior of Nonlinear Autonomous Systems
  • 5.3.1. Finding the Equilibrium Points
  • 5.3.2. Determining the Type of an Equilibrium
  • 5.3.3. A Limit Cycle—the Van der Pol Equation
  • 5.4. Bifurcations for Systems
  • 5.4.1. Bifurcation in a Spring-mass Model
  • 5.4.2. Bifurcation of a Predator-prey Model
  • 5.4.3. Bifurcation Analysis Applied to a Competing Species Model
  • 5.5. Student Projects
  • 5.5.1. The Wilson-Cowan Equations
  • 5.5.2. A New Predator-prey Equation—Putting It All Together
  • Chapter 6. Laplace Transforms
  • 6.1. Definition and Some Simple Laplace Transforms
  • 6.1.1. Four Simple Laplace Transforms
  • 6.1.2. Linearity of the Laplace Transform
  • 6.1.3. Transforming the Derivative of 𝑓(𝑡)
  • 6.2. Solving Equations, the Inverse Laplace Transform
  • 6.2.1. Partial Fraction Expansions
  • 6.3. Extending the Table
  • 6.3.1. Inverting a Term with an Irreducible Quadratic Denominator
  • 6.3.2. Solving Linear Systems with Laplace Transforms
  • 6.4. The Unit Step Function
  • 6.5. Convolution and the Impulse Function
  • 6.5.1. The Convolution Integral
  • 6.5.2. The Impulse Function
  • 6.5.3. Impulse Response of a Linear, Time-invariant System
  • Chapter 7. Introduction to Partial Differential Equations
  • 7.1. Solving Partial Differential Equations
  • 7.1.1. An Overview of the Method of Separation of Variables
  • 7.2. Orthogonal Functions and Trigonometric Fourier Series
  • 7.2.1. Orthogonal Families of Functions
  • 7.2.2. Properties of Fourier Series, Cosine and Sine Series
  • 7.3. Boundary-Value Problems: Sturm-Liouville otoc { } Equations
  • Chapter 8. Solving Second-order Partial Differential Equations
  • 8.1. Classification of Linear Second-order Partial Differential Equations
  • 8.2. The 1-dimensional Heat Equation
  • 8.2.1. Solution of the Heat Equation by Separation of Variables
  • 8.2.2. Other Boundary Conditions for the Heat Equation
  • 8.3. The 1-dimensional Wave Equation
  • 8.3.1. Solution of the Wave Equation by Separation of Variables
  • 8.3.2. D’Alembert’s Solution of the Wave Equation on an Infinite Interval
  • 8.4. Numerical Solution of Parabolic and Hyperbolic Equations
  • 8.5. Laplace’s Equation
  • 8.6. Student Project: Harvested Diffusive Logistic Equation
  • Appendix
  • Appendix A. Answers to Odd-numbered Exercises
  • Appendix B. Derivative and Integral Formulas
  • Appendix C. Cofactor Method for Determinants
  • Appendix D. Cramer’s Rule for Solving Systems of Linear Equations
  • Appendix E. The Wronskian
  • Appendix F. Table of Laplace Transforms
  • Appendix G. Review of Partial Derivatives
  • Index
  • Back Cover
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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