Differential Equations : An Applied Approach

by Cushing, J. M.

• ISBN13: 9780130449306
• ISBN10: 013044930X
• Edition: 1st
• Format: Hardcover
• Copyright: 2004-01-01
• Publisher: Addison Wesley

Summary

This foundational introduction to modern dynamical systems covers traditional subjects, as well as modern topics such as fundamentals of dynamical systems theory and bifurcation theory. The volume emphasizes analyzing solutions rather than finding solution formulas, introduces numerical methods early in the book and provides case studies for each subject area. Many applications are accompanied with real data and are drawn from many disciplines including many from biological subjects. The volume addresses first order equations, linear first order equations, nonlinear first order equations, systems and higher order equations, homogeneous linear systems and higher order equations, nonhomogeneous linear systems, approximations and series solutions, nonlinear systems and Laplace transforms. For those needing an introduction to modern dynamical systems.

Table of Contents

Preface	xi
Introduction	1	(18)
1 Preliminaries	1	(8)
2 Mathematical Models	9	(10)
1 First-Order Equations	19	(26)
1.1 The Fundamental Existence Theorem	20	(6)
1.2 Approximation of Solutions	26	(11)
1.3 Another Numerical Algorithm	37	(3)
1.4 Chapter Summary and Exercises	40	(3)
1.5 Applications	43	(2)
2 Linear First-Order Equations
2.1 The Solution of Linear Equations	45	(10)
2.2 Properties of Solutions	55	(5)
2.3 The Method of Undetermined Coefficients	60	(5)
2.4 Autonomous Linear Equations	65	(4)
2.5 Chapter Summary and Exercises	69	(3)
2.6 Applications	72	(1)
3 Nonlinear First-Order Equations	73	(63)
3.1 Autonomous Equations	74	(28)
3.2 Separable Equations	102	(10)
3.3 Change of Variables	112	(4)
3.4 Approximation Formulas	116	(16)
3.5 Chapter Summary and Exercises	132	(3)
3.6 Applications	135	(1)
4 Systems and Higher-Order Equations	136	(40)
4.1 The Fundamental Existence Theorem	138	(4)
4.2 Approximating Solutions of Systems	142	(11)
4.3 Linear Systems of Equations	153	(13)
4.4 Matrix Notation	166	(6)
4.5 Chapter Summary and Exercises	172	(3)
4.6 Applications	175	(1)
5 Homogeneous Linear Systems and Higher-Order Equations	176	(50)
5.1 Introduction	176	(4)
5.2 Homogeneous Systems with Constant Coefficients	180	(12)
5.3 Homogeneous Second-Order Equations	192	(4)
5.4 Phase Plane Portraits	196	(17)
5.5 Matrices and Eigenvalues	213	(5)
5.6 Systems of Three or More Equations	218	(6)
5.7 Chapter Summary and Exercises	224	(1)
5.8 Applications	225	(1)
6 Nonhomogeneous Linear Systems	226	(31)
6.1 The Method of Undetermined Coefficients	228	(7)
6.2 The Variation of Constants Formula	235	(10)
6.3 Matrix Notation	245	(9)
6.4 Chapter Summary and Exercises	254	(2)
6.5 Applications	256	(1)
7 Approximations and Series Solutions	257	(28)
7.1 Taylor Polynomials and Picard Iterates	257	(11)
7.2 A Perturbation Method	268	(8)
7.3 Power Series Solutions	276	(7)
7.4 Chapter Summary and Exercises	283	(1)
7.5 Applications	284	(1)
8 Nonlinear Systems	285	(46)
8.1 Introduction	285	(2)
8.2 Phase Plane Portraits and Equilibria	287	(5)
8.3 The Linearization Principle	292	(6)
8.4 Local Phase Plane Portraits	298	(6)
8.5 Global Phase Plane Portraits	304	(7)
8.6 Bifurcations	311	(9)
8.7 Higher-Dimensional Systems	320	(8)
8.8 Chapter Summary and Exercises	328	(2)
8.9 Applications	330	(1)
9 Laplace Transforms	331	(41)
9.1 Introduction	331	(8)
9.2 The Laplace Transform	339	(6)
9.3 Linearity and the Inverse Laplace Transform	345	(3)
9.4 Properties of the Laplace Transform	348	(11)
9.5 Solution of Initial Value Problems	359	(10)
9.6 Chapter Summary and Exercises	369	(2)
9.7 Applications	371	(1)
Answers to Selected Exercises	372
Index	I-1

Excerpts

This book is an outgrowth of lecture notes I wrote for an introductory course on differential equations and modeling that I have taught at the University of Arizona for over twenty years. The book offers a blend of topics traditionally found in a first course on differential equations with a coherent selection of applied and contemporary topics that are of interest to a growing and diversifying audience in science and engineering. These topics are supplemented with a brief introduction to mathematical modeling and many applications and in-depth case studies (often involving real data). 'There is enough material and flexibility in the book that an instructor can design a course with any of several different emphases. For example, by appropriate choices of topics one can devise a reasonably traditional course that focuses on algebraic and calculus methods, solution formula techniques, etc.; or a course that has a dynamical systems emphasis centered on asymptotic dynamics, stability analysis, bifurcation theory, etc.; or a course with a significant component of mathematical modeling, applications, and case studies. In my own teaching I strive to strike a balance among these various themes. To do this is sometimes a difficult task, and the balance at which I arrive usually varies from semester to semester, primarily in response to the backgrounds, interests, and needs of my students. Students in my classes have, over the years, come from virtually every college on our campus: sciences, engineering, agriculture, education, business, and even the fine arts. My classes typically include students majoring in mathematics, mathematics education, engineering, physics, chemistry, and various fields of biology. It is not intended, of course, that all material in the book be covered in a single course. At some schools, some topics in the book might be covered in prerequisite courses and some topics might be taught in other courses. For example, it is now often the case that slope fields, the Euler numerical algorithm, and the separation of variables method for single first-order equations are taught in a first course on calculus. This is the case at the University of Arizona, and therefore I treat these topics as review material. Laplace transforms are not taught in introductory differential courses at the University of Arizona (where they are taught in mathematical methods courses for scientists and engineers), but I do include Laplace transforms in the text because they are an important topic at many schools. All topics in the text are presented in a self contained way and the book can be successfully used with only a traditional first course in (single variable) calculus as a prerequisite. At some schools linear algebra is prerequisite for a first course in differential equations (or students have had at least some exposure to matrix algebra), while at other schools this is not the case. At yet other schools first courses in linear algebra and differential equations are co-requisite and might even be taught in the same course. A first course in linear algebra is a prerequisite for my course at the University of Arizona. The book, however, develops all topics first without a linear or matrix algebra prerequisite and include follow-up sections that introduce the use of these subjects at appropriate times. I personally have found that my students who have previously studied linear and matrix algebra benefit nonetheless from an introductory presentation without use of these topics, succeeded by a follow-up that brings to play relevant matrix and linear algebraic topics. For a course without a matrix or linear algebra prerequisite, the follow-up sections can simply be omitted, or be used as a brief introduction of these topics (matrix notation and algebra, eigenvalues and eigenvectors, etc.). A significant component of the course I teach at the University of Arizona consists of applications and case studies. Again, the is