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Differential Equations: Techniques, Theory, and Applications

Barbara D. MacCluer University of Virginia, Charlottesville, VA
Paul S. Bourdon University of Virginia, Charlottesville, VA
Thomas L. Kriete University of Virginia, Charlottesville, VA
Available Formats:
Hardcover ISBN: 978-1-4704-4797-7
Product Code: MBK/125
List Price: $125.00 MAA Member Price:$112.50
AMS Member Price: $100.00 Electronic ISBN: 978-1-4704-5438-8 Product Code: MBK/125.E List Price:$125.00
MAA Member Price: $112.50 AMS Member Price:$100.00
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List Price: $187.50 MAA Member Price:$168.75
AMS Member Price: $150.00 Click above image for expanded view Differential Equations: Techniques, Theory, and Applications Barbara D. MacCluer University of Virginia, Charlottesville, VA Paul S. Bourdon University of Virginia, Charlottesville, VA Thomas L. Kriete University of Virginia, Charlottesville, VA Available Formats:  Hardcover ISBN: 978-1-4704-4797-7 Product Code: MBK/125  List Price:$125.00 MAA Member Price: $112.50 AMS Member Price:$100.00
 Electronic ISBN: 978-1-4704-5438-8 Product Code: MBK/125.E
 List Price: $125.00 MAA Member Price:$112.50 AMS Member Price: $100.00 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:$187.50 MAA Member Price: $168.75 AMS Member Price:$150.00
• Book Details

2019; 874 pp
MSC: 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 large-scale projects. In-depth 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.

• 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, a leading provider of online instructional tools for both faculty and students.
• A student solution manual is also available for purchase here.
• A preview of Chapter 10 is available here.
• An index of the Projects offered for this title is available here.

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. First-Order 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 Runge-Kutta Algorithms
• 3.3. Optical illusions and other applications
• Chapter 4. Higher-Order Linear Homogeneous Equations
• 4.1. Introduction to second-order equations
• 4.2. Linear operators
• 4.3. Linear independence
• 4.4. Constant coefficient second-order equations
• 4.5. Repeated roots and reduction of order
• 4.6. Higher-order equations
• 4.7. Higher-order constant coefficient equations
• 4.8. Modeling with second-order equations
• Chapter 5. Higher-Order 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
• 9.3. The matrix exponential, Part II
• 9.4. Nonhomogeneous constant coefficient systems
• 9.5. Periodic forcing and the steady-state 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
• 11.5. 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
• Bibliography
• Index
• Back Cover

• Reviews

• ...this book provides a self-contained 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 University-level teaching of its subject.

Nikola Popović, University of Edinburgh
• 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
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
2019; 874 pp
MSC: 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 large-scale projects. In-depth 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.

• 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, a leading provider of online instructional tools for both faculty and students.
• A student solution manual is also available for purchase here.
• A preview of Chapter 10 is available here.
• An index of the Projects offered for this title is available here.

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. First-Order 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 Runge-Kutta Algorithms
• 3.3. Optical illusions and other applications
• Chapter 4. Higher-Order Linear Homogeneous Equations
• 4.1. Introduction to second-order equations
• 4.2. Linear operators
• 4.3. Linear independence
• 4.4. Constant coefficient second-order equations
• 4.5. Repeated roots and reduction of order
• 4.6. Higher-order equations
• 4.7. Higher-order constant coefficient equations
• 4.8. Modeling with second-order equations
• Chapter 5. Higher-Order 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
• 9.3. The matrix exponential, Part II
• 9.4. Nonhomogeneous constant coefficient systems
• 9.5. Periodic forcing and the steady-state 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
• 11.5. 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