Softcover ISBN:  9781470463298 
Product Code:  TEXT/43.S 
List Price:  $75.00 
MAA Member Price:  $56.25 
AMS Member Price:  $56.25 
Electronic ISBN:  9781470451080 
Product Code:  TEXT/43.E 
List Price:  $75.00 
MAA Member Price:  $56.25 
AMS Member Price:  $56.25 

Book DetailsAMS/MAA TextbooksVolume: 43; 2019; 402 ppMSC: Primary 34; 35;
A thoroughly modern textbook for the sophomorelevel 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 flippedclassroom pedagogical approach or for selfstudy for an advanced undergraduate or beginning graduate student.
This second edition of Noonburg's bestselling textbook includes two new chapters on partial differential equations, making the book usable for a twosemester 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 authordesigned 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.ReadershipUndergraduate 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, InitialValue Problems

1.3. Modeling with Differential Equations

Chapter 2. Firstorder Differential Equations

2.1. Separable Firstorder 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 Firstorder Differential Equations

2.3.1. Application: Singlecompartment Mixing Problem

2.4. Existence and Uniqueness of Solutions

2.5. More Analytic Methods for Nonlinear Firstorder 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. Fourthorder RungeKutta Method

2.7. Autonomous Equations, the Phase Line

2.7.1. Stability—Sinks, Sources, and Nodes

Bifurcation in Equations with Parameters

Chapter 3. Secondorder Differential Equations

3.1. General Theory of Homogeneous Linear Equations

3.2. Homogeneous Linear Equations with Constant Coefficients

3.2.1. Secondorder Equation with Constant Coefficients

3.2.2. Equations of Order Greater Than Two

3.3. The Springmass Equation

3.3.1. Derivation of the Springmass Equation

3.3.2. The Unforced Springmass System

3.4. Nonhomogeneous Linear Equations

3.4.1. Method of Undetermined Coefficients

3.4.2. Variation of Parameters

3.5. The Forced Springmass System

Beats and Resonance

3.6. Linear Secondorder Equations with Nonconstant Coefficients

3.6.1. The CauchyEuler Equation

3.6.2. Series Solutions

3.7. Autonomous Secondorder Differential Equations

3.7.1. Numerical Methods

3.7.2. Autonomous Equations and the Phase Plane

Chapter 4. Linear Systems ofFirstorder Differential Equations

4.1. Introduction to Systems

4.1.1. Writing Differential Equations as a Firstorder 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 Springmass 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 Tracedeterminant 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 Springmass Model

5.4.2. Bifurcation of a Predatorprey Model

5.4.3. Bifurcation Analysis Applied to a Competing Species Model

5.5. Student Projects

5.5.1. The WilsonCowan Equations

5.5.2. A New Predatorprey 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, Timeinvariant 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. BoundaryValue Problems: SturmLiouville otoc { } Equations

Chapter 8. Solving Secondorder Partial Differential Equations

8.1. Classification of Linear Secondorder Partial Differential Equations

8.2. The 1dimensional 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 1dimensional 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 Oddnumbered 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


Additional Material

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A thoroughly modern textbook for the sophomorelevel 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 flippedclassroom pedagogical approach or for selfstudy for an advanced undergraduate or beginning graduate student.
This second edition of Noonburg's bestselling textbook includes two new chapters on partial differential equations, making the book usable for a twosemester 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 authordesigned 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

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, InitialValue Problems

1.3. Modeling with Differential Equations

Chapter 2. Firstorder Differential Equations

2.1. Separable Firstorder 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 Firstorder Differential Equations

2.3.1. Application: Singlecompartment Mixing Problem

2.4. Existence and Uniqueness of Solutions

2.5. More Analytic Methods for Nonlinear Firstorder 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. Fourthorder RungeKutta Method

2.7. Autonomous Equations, the Phase Line

2.7.1. Stability—Sinks, Sources, and Nodes

Bifurcation in Equations with Parameters

Chapter 3. Secondorder Differential Equations

3.1. General Theory of Homogeneous Linear Equations

3.2. Homogeneous Linear Equations with Constant Coefficients

3.2.1. Secondorder Equation with Constant Coefficients

3.2.2. Equations of Order Greater Than Two

3.3. The Springmass Equation

3.3.1. Derivation of the Springmass Equation

3.3.2. The Unforced Springmass System

3.4. Nonhomogeneous Linear Equations

3.4.1. Method of Undetermined Coefficients

3.4.2. Variation of Parameters

3.5. The Forced Springmass System

Beats and Resonance

3.6. Linear Secondorder Equations with Nonconstant Coefficients

3.6.1. The CauchyEuler Equation

3.6.2. Series Solutions

3.7. Autonomous Secondorder Differential Equations

3.7.1. Numerical Methods

3.7.2. Autonomous Equations and the Phase Plane

Chapter 4. Linear Systems ofFirstorder Differential Equations

4.1. Introduction to Systems

4.1.1. Writing Differential Equations as a Firstorder 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 Springmass 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 Tracedeterminant 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 Springmass Model

5.4.2. Bifurcation of a Predatorprey Model

5.4.3. Bifurcation Analysis Applied to a Competing Species Model

5.5. Student Projects

5.5.1. The WilsonCowan Equations

5.5.2. A New Predatorprey 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, Timeinvariant 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. BoundaryValue Problems: SturmLiouville otoc { } Equations

Chapter 8. Solving Secondorder Partial Differential Equations

8.1. Classification of Linear Secondorder Partial Differential Equations

8.2. The 1dimensional 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 1dimensional 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 Oddnumbered 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