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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
Available Formats:
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 Electronic ISBN: 978-1-4704-5108-0 Product Code: TEXT/43.E List Price:$75.00
MAA Member Price: $56.25 AMS Member Price:$56.25
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List Price: $112.50 MAA Member Price:$84.38
AMS Member Price: $84.38 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 Available Formats:  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
 Electronic ISBN: 978-1-4704-5108-0 Product Code: TEXT/43.E
 List Price: $75.00 MAA Member Price:$56.25 AMS Member Price: $56.25 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$112.50 MAA Member Price: $84.38 AMS Member Price:$84.38
• Book Details

AMS/MAA Textbooks
Volume: 432019; 402 pp
MSC: Primary 34; 35;

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.

There exists a number of author-designed extensive student projects and a collection of additional exercises on the book's webpage. Click on "Supplemental Materials" on the lower left of this page to access them.

An instructor's manual for this title is available electronically to those instructors who have adopted the textbook for classroom use. Please send email to textbooks@ams.org for more information.

Online assignments for this title are available in WebAssign. WebAssign is a leading provider of online instructional tools for both faculty and students.

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

• Title page
• 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 reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
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;

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.

There exists a number of author-designed extensive student projects and a collection of additional exercises on the book's webpage. Click on "Supplemental Materials" on the lower left of this page to access them.

An instructor's manual for this title is available electronically to those instructors who have adopted the textbook for classroom use. Please send email to textbooks@ams.org for more information.

Online assignments for this title are available in WebAssign. WebAssign is a leading provider of online instructional tools for both faculty and students.

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

• Title page
• 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 reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
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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