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Hardcover ISBN:  9781470447977 
Product Code:  MBK/125 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470454388 
Product Code:  MBK/125.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470447977 
eBook ISBN:  9781470454388 
Product Code:  MBK/125.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book Details2019; 874 ppMSC: Primary 34; 35; Secondary 97
Differential Equations: Techniques, Theory, and Applications is designed for a modern first course in differential equations either one or two semesters in length. The organization of the book interweaves the three components in the subtitle, with each building on and supporting the others. Techniques include not just computational methods for producing solutions to differential equations, but also qualitative methods for extracting conceptual information about differential equations and the systems modeled by them. Theory is developed as a means of organizing, understanding, and codifying general principles. Applications show the usefulness of the subject as a whole and heighten interest in both solution techniques and theory. Formal proofs are included in cases where they enhance core understanding; otherwise, they are replaced by informal justifications containing key ideas of a proof in a more conversational format. Applications are drawn from a wide variety of fields: those in physical science and engineering are prominent, of course, but models from biology, medicine, ecology, economics, and sports are also featured.
The 1,400+ exercises are especially compelling. They range from routine calculations to largescale projects. Indepth student projects, many with Mathematica files, are available here as Supplemental Material. The more difficult problems, both theoretical and applied, are typically presented in manageable steps. The hundreds of meticulously detailed modeling problems were deliberately designed along pedagogical principles found especially effective in the MAA study Characteristics of Successful Calculus Programs, namely, that asking students to work problems that require them to grapple with concepts (or even proofs) and do modeling activities is key to successful student experiences and retention in STEM programs. The exposition itself is exceptionally readable, rigorous yet conversational. Students will find it inviting and approachable. The text supports many different styles of pedagogy from traditional lecture to a flipped classroom model. The availability of a computer algebra system is not assumed, but there are many opportunities to incorporate the use of one.
Ancillaries:
ReadershipUndergraduate students interested in differential equations.

Table of Contents

Cover

Title page

Preface

Chapter 1. Introduction

1.1. What is a differential equation?

1.2. What is a solution?

1.3. More on direction fields: Isoclines

Chapter 2. FirstOrder Equations

2.1. Linear equations

2.2. Separable equations

2.3. Applications: Time of death, time at depth, and ancient timekeeping

2.4. Existence and uniqueness theorems

2.5. Population and financial models

2.6. Qualitative solutions of autonomous equations

2.7. Change of variable

2.8. Exact equations

Chapter 3. Numerical Methods

3.1. Euler’s method

3.2. Improving Euler’s method: The Heun and RungeKutta Algorithms

3.3. Optical illusions and other applications

Chapter 4. HigherOrder Linear Homogeneous Equations

4.1. Introduction to secondorder equations

4.2. Linear operators

4.3. Linear independence

4.4. Constant coefficient secondorder equations

4.5. Repeated roots and reduction of order

4.6. Higherorder equations

4.7. Higherorder constant coefficient equations

4.8. Modeling with secondorder equations

Chapter 5. HigherOrder Linear Nonhomogeneous Equations

5.1. Introduction to nonhomogeneous equations

5.2. Annihilating operators

5.3. Applications of nonhomogeneous equations

5.4. Electric circuits

Chapter 6. Laplace Transforms

6.1. Laplace transforms

6.2. The inverse Laplace transform

6.3. Solving initial value problems with Laplace transforms

6.4. Applications

6.5. Laplace transforms, simple systems, and Iwo Jima

6.6. Convolutions

6.7. The delta function

Chapter 7. Power Series Solutions

7.1. Motivation for the study of power series solutions

7.2. Review of power series

7.3. Series solutions

7.4. Nonpolynomial coefficients

7.5. Regular singular points

7.6. Bessel’s equation

Chapter 8. Linear Systems I

8.1. Nelson at Trafalgar and phase portraits

8.2. Vectors, vector fields, and matrices

8.3. Eigenvalues and eigenvectors

8.4. Solving linear systems

8.5. Phase portraits via ray solutions

8.6. More on phase portraits: Saddle points and nodes

8.7. Complex and repeated eigenvalues

8.8. Applications: Compartment models

8.9. Classifying equilibrium points

Chapter 9. Linear Systems II

9.1. The matrix exponential, Part I

9.2. A return to the Existence and Uniqueness Theorem

9.3. The matrix exponential, Part II

9.4. Nonhomogeneous constant coefficient systems

9.5. Periodic forcing and the steadystate solution

Chapter 10. Nonlinear Systems

10.1. Introduction: Darwin’s finches

10.2. Linear approximation: The major cases

10.3. Linear approximation: The borderline cases

10.4. More on interacting populations

10.5. Modeling the spread of disease

10.6. Hamiltonians, gradient systems, and Lyapunov functions

10.7. Pendulums

10.8. Cycles and limit cycles

Chapter 11. Partial Differential Equations and Fourier Series

11.1. Introduction: Three interesting partial differential equations

11.2. Boundary value problems

11.3. Partial differential equations: A first look

11.4. Advection and diffusion

11.5. Functions as vectors

11.6. Fourier series

11.7. The heat equation

11.8. The wave equation: Separation of variables

11.9. The wave equation: D’Alembert’s method

11.10. Laplace’s equation

Notes and Further Reading

Selected Answers to Exercises

Bibliography

Index

Back Cover


Additional Material

Reviews

...this book provides a selfcontained and complete introduction to differential equations that does justice to a field which has steadily grown in importance in recent years. Given that it has been written with the student learning experience firmly in mind, and due to its innovative take on what is, in essence, core syllabus for any standard introductory course on differential equations, it certainly represents a welcome and valuable resource for Universitylevel teaching of its subject.
Nikola Popović, University of Edinburgh


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 Book Details
 Table of Contents
 Additional Material
 Reviews
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Differential Equations: Techniques, Theory, and Applications is designed for a modern first course in differential equations either one or two semesters in length. The organization of the book interweaves the three components in the subtitle, with each building on and supporting the others. Techniques include not just computational methods for producing solutions to differential equations, but also qualitative methods for extracting conceptual information about differential equations and the systems modeled by them. Theory is developed as a means of organizing, understanding, and codifying general principles. Applications show the usefulness of the subject as a whole and heighten interest in both solution techniques and theory. Formal proofs are included in cases where they enhance core understanding; otherwise, they are replaced by informal justifications containing key ideas of a proof in a more conversational format. Applications are drawn from a wide variety of fields: those in physical science and engineering are prominent, of course, but models from biology, medicine, ecology, economics, and sports are also featured.
The 1,400+ exercises are especially compelling. They range from routine calculations to largescale projects. Indepth student projects, many with Mathematica files, are available here as Supplemental Material. The more difficult problems, both theoretical and applied, are typically presented in manageable steps. The hundreds of meticulously detailed modeling problems were deliberately designed along pedagogical principles found especially effective in the MAA study Characteristics of Successful Calculus Programs, namely, that asking students to work problems that require them to grapple with concepts (or even proofs) and do modeling activities is key to successful student experiences and retention in STEM programs. The exposition itself is exceptionally readable, rigorous yet conversational. Students will find it inviting and approachable. The text supports many different styles of pedagogy from traditional lecture to a flipped classroom model. The availability of a computer algebra system is not assumed, but there are many opportunities to incorporate the use of one.
Ancillaries:
Undergraduate students interested in differential equations.

Cover

Title page

Preface

Chapter 1. Introduction

1.1. What is a differential equation?

1.2. What is a solution?

1.3. More on direction fields: Isoclines

Chapter 2. FirstOrder Equations

2.1. Linear equations

2.2. Separable equations

2.3. Applications: Time of death, time at depth, and ancient timekeeping

2.4. Existence and uniqueness theorems

2.5. Population and financial models

2.6. Qualitative solutions of autonomous equations

2.7. Change of variable

2.8. Exact equations

Chapter 3. Numerical Methods

3.1. Euler’s method

3.2. Improving Euler’s method: The Heun and RungeKutta Algorithms

3.3. Optical illusions and other applications

Chapter 4. HigherOrder Linear Homogeneous Equations

4.1. Introduction to secondorder equations

4.2. Linear operators

4.3. Linear independence

4.4. Constant coefficient secondorder equations

4.5. Repeated roots and reduction of order

4.6. Higherorder equations

4.7. Higherorder constant coefficient equations

4.8. Modeling with secondorder equations

Chapter 5. HigherOrder Linear Nonhomogeneous Equations

5.1. Introduction to nonhomogeneous equations

5.2. Annihilating operators

5.3. Applications of nonhomogeneous equations

5.4. Electric circuits

Chapter 6. Laplace Transforms

6.1. Laplace transforms

6.2. The inverse Laplace transform

6.3. Solving initial value problems with Laplace transforms

6.4. Applications

6.5. Laplace transforms, simple systems, and Iwo Jima

6.6. Convolutions

6.7. The delta function

Chapter 7. Power Series Solutions

7.1. Motivation for the study of power series solutions

7.2. Review of power series

7.3. Series solutions

7.4. Nonpolynomial coefficients

7.5. Regular singular points

7.6. Bessel’s equation

Chapter 8. Linear Systems I

8.1. Nelson at Trafalgar and phase portraits

8.2. Vectors, vector fields, and matrices

8.3. Eigenvalues and eigenvectors

8.4. Solving linear systems

8.5. Phase portraits via ray solutions

8.6. More on phase portraits: Saddle points and nodes

8.7. Complex and repeated eigenvalues

8.8. Applications: Compartment models

8.9. Classifying equilibrium points

Chapter 9. Linear Systems II

9.1. The matrix exponential, Part I

9.2. A return to the Existence and Uniqueness Theorem

9.3. The matrix exponential, Part II

9.4. Nonhomogeneous constant coefficient systems

9.5. Periodic forcing and the steadystate solution

Chapter 10. Nonlinear Systems

10.1. Introduction: Darwin’s finches

10.2. Linear approximation: The major cases

10.3. Linear approximation: The borderline cases

10.4. More on interacting populations

10.5. Modeling the spread of disease

10.6. Hamiltonians, gradient systems, and Lyapunov functions

10.7. Pendulums

10.8. Cycles and limit cycles

Chapter 11. Partial Differential Equations and Fourier Series

11.1. Introduction: Three interesting partial differential equations

11.2. Boundary value problems

11.3. Partial differential equations: A first look

11.4. Advection and diffusion

11.5. Functions as vectors

11.6. Fourier series

11.7. The heat equation

11.8. The wave equation: Separation of variables

11.9. The wave equation: D’Alembert’s method

11.10. Laplace’s equation

Notes and Further Reading

Selected Answers to Exercises

Bibliography

Index

Back Cover

...this book provides a selfcontained and complete introduction to differential equations that does justice to a field which has steadily grown in importance in recent years. Given that it has been written with the student learning experience firmly in mind, and due to its innovative take on what is, in essence, core syllabus for any standard introductory course on differential equations, it certainly represents a welcome and valuable resource for Universitylevel teaching of its subject.
Nikola Popović, University of Edinburgh