MSC: Primary 34; 35; Secondary 97
Print ISBN: 978-1-4704-4797-7
Product Code: MBK/125
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Electronic ISBN: 978-1-4704-5438-8
Product Code: MBK/125.E
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AMS Member Price: $100.00
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Differential Equations: Techniques, Theory, and Applications
Share this pageBarbara D. MacCluer; Paul S. Bourdon; Thomas L. Kriete
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. 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.
FREELY DOWNLOADABLE CONTENT RELATED TO THE COVID-19 PANDEMIC:
—
Chapter
10, covering Nonlinear Systems, with a section on modeling the spread
of disease.
— A
project
(Zip, 221kb) on the Transmissibility of COVID-19, with exercises and a
starter Mathematica file.
Readership
Undergraduate students interested in differential equations.
Table of Contents
Table of Contents
Differential Equations: Techniques, Theory, and Applications
- Cover Cover11
- Title page iii4
- Preface ix10
- Chapter 1. Introduction 116
- Chapter 2. First-Order Equations 2540
- 2.1. Linear equations 2540
- 2.2. Separable equations 3550
- 2.3. Applications: Time of death, time at depth, and ancient timekeeping 4560
- 2.4. Existence and uniqueness theorems 6580
- 2.5. Population and financial models 8398
- 2.6. Qualitative solutions of autonomous equations 98113
- 2.7. Change of variable 112127
- 2.8. Exact equations 121136
- Chapter 3. Numerical Methods 139154
- Chapter 4. Higher-Order Linear Homogeneous Equations 171186
- 4.1. Introduction to second-order equations 171186
- 4.2. Linear operators 192207
- 4.3. Linear independence 209224
- 4.4. Constant coefficient second-order equations 216231
- 4.5. Repeated roots and reduction of order 228243
- 4.6. Higher-order equations 240255
- 4.7. Higher-order constant coefficient equations 245260
- 4.8. Modeling with second-order equations 254269
- Chapter 5. Higher-Order Linear Nonhomogeneous Equations 265280
- Chapter 6. Laplace Transforms 319334
- Chapter 7. Power Series Solutions 381396
- Chapter 8. Linear Systems I 443458
- 8.1. Nelson at Trafalgar and phase portraits 443458
- 8.2. Vectors, vector fields, and matrices 457472
- 8.3. Eigenvalues and eigenvectors 472487
- 8.4. Solving linear systems 482497
- 8.5. Phase portraits via ray solutions 495510
- 8.6. More on phase portraits: Saddle points and nodes 507522
- 8.7. Complex and repeated eigenvalues 524539
- 8.8. Applications: Compartment models 536551
- 8.9. Classifying equilibrium points 549564
- Chapter 9. Linear Systems II 563578
- Chapter 10. Nonlinear Systems 615630
- 10.1. Introduction: Darwin’s finches 615630
- 10.2. Linear approximation: The major cases 627642
- 10.3. Linear approximation: The borderline cases 647662
- 10.4. More on interacting populations 653668
- 10.5. Modeling the spread of disease 668683
- 10.6. Hamiltonians, gradient systems, and Lyapunov functions 683698
- 10.7. Pendulums 699714
- 10.8. Cycles and limit cycles 708723
- Chapter 11. Partial Differential Equations and Fourier Series 717732
- 11.1. Introduction: Three interesting partial differential equations 717732
- 11.2. Boundary value problems 719734
- 11.3. Partial differential equations: A first look 727742
- 11.4. Advection and diffusion 734749
- 11.5. Functions as vectors 745760
- 11.6. Fourier series 760775
- 11.7. The heat equation 777792
- 11.8. The wave equation: Separation of variables 792807
- 11.9. The wave equation: D’Alembert’s method 804819
- 11.10. Laplace’s equation 812827
- Notes and Further Reading 833848
- Selected Answers to Exercises 837852
- Bibliography 863878
- Index 867882
- Back Cover Back Cover1890