Differential Equations with Boundary Value Problems

by Hollis, Selwyn

• ISBN13: 9780130159274
• ISBN10: 0130159271
• Edition: 1st
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: Pearson College Div

Summary

This book provides readers with a solid introduction to differential equations and their applications emphasizing analytical, qualitative, and numerical methods. Numerical methods are presented early in the text, including a discussion of error estimates for the Euler, Heun, and Runge-Kutta methods. Systems and the phase plane are also introduced early, first in the context of pairs first-order equations, and then in the context of second-order linear equations. Other chapter topics include the Laplace transform, linear first-order systems, geometry of autonomous systems in the plane, nonlinear systems in applications, diffusion problems and Fourier series, and further topics in PDEs.

Table of Contents

Preface	ix
Introduction	1	(24)
Prologue: What Are Differential Equations?	1	(3)
Four Introductory Models	4	(10)
Fundamental Concepts and Terminology	14	(11)
Linear First-Order Equations	25	(26)
Methods of Solution	25	(7)
Some Elementary Applications	32	(12)
Projectile Motion with Resistance	33	(4)
Mixing Problems	37	(4)
Circuits	41	(3)
Generalized Solutions	44	(7)
Nonlinear First-Order Equations I	51	(36)
Direction Fields and Numerical Approximation	51	(8)
Separable Equations	59	(5)
Bernoulli and Riccati Equations	64	(4)
Reduction of Order	68	(5)
Nonlinear First-order Equations in Applications	73	(14)
Motion with Nonlinear Resistance	73	(5)
Torricelli's Law	78	(6)
Nonlinear Circuits	84	(3)
Nonlinear First-Order Equations II	87	(46)
Construction of Local Solutions	87	(6)
Existence and Uniqueness	93	(6)
Qualitative and Asymptotic Behavior	99	(6)
The Logistic Population Model	105	(10)
Numerical Methods	115	(10)
A First Look at Systems	125	(8)
Second-Order Linear Equations I	133	(52)
Introduction: Modeling Vibrations	133	(6)
State Variables and Numerical Approximation	139	(8)
Operators and Linearity	147	(5)
Solutions and Linear Independence	152	(10)
Variation of Constants and Green's Functions	162	(4)
Power-Series Solutions	166	(13)
Polynomial Solutions	179	(6)
Second-Order Linear Equations II	185	(35)
Homogeneous Equations with Constant Coefficients	185	(6)
Exponential Shift	191	(5)
Complex Roots	196	(6)
Real Solutions from Complex Solutions	202	(4)
Unforced Vibrations	206	(5)
Periodic Force and Response	211	(9)
The Laplace Transform	220	(39)
Definition and Basic Properties	220	(7)
More Transforms and Further Properties	227	(7)
Heaviside Functions and Piecewise-Defined Inputs	234	(7)
Periodic Inputs	241	(6)
Impulses and the Dirac Distribution	247	(5)
Convolution	252	(7)
First-Order Linear Systems	259	(50)
Introduction	259	(6)
Two Ad Hoc Methods	265	(4)
Vector-Valued Functions and Linear Independence	269	(7)
Evolution Matrices and Variation of Constants	276	(7)
Autonomous Systems: Eigenvalues and Eigenvectors	283	(10)
eAt and the Cayley-Hamilton Therorem	293	(8)
Asymptotic Stability	301	(8)
Geometry of Autonomous Systems in the Plane	309	(57)
The Phase Plane	309	(11)
Phase Portraits of Homogeneous Linear Systems	320	(14)
Phase Portraits of Nonlinear Systems	334	(10)
Limit Cycles	344	(11)
Beyond the Plane	355	(11)
Nonlinear Systems in Applications	366	(42)
Lotka-Volterra Systems in Ecology	366	(14)
Infectious Disease and Epidemics	380	(5)
Other Biological Models	385	(7)
Chemical Systems	392	(7)
Mechanics	399	(9)
Diffusion Problems and Fourier Series	408	(64)
The Basic Diffusion Problem	408	(9)
Solutions by Separation of Variables	417	(10)
Fourier Series	427	(10)
Fourier Sine and Cosine Series	437	(8)
Sturm-Liouville Eigenvalue Problems	445	(13)
Singular Sturm-Liouville Problems	458	(4)
Eigenfunction Expansions	462	(10)
Further Topics in PDEs	472	(40)
The Wave Equation	472	(12)
The 2-D Laplace Equation	484	(13)
The 2-D Diffusion Equation	497	(15)
Appendices	512	(119)
I Linear Algebra	512	(45)
I.A. Linear Algebraic Equations	512	(11)
I.B. Matrices	523	(20)
I.C. Eigenvalues and Eigenvectors	543	(14)
II Continuity and Differentiability on Intervals	557	(5)
III Local Existence and Uniqueness Proofs	562	(15)
IV Linear Differential Equations of Arbitrary Order	577	(14)
V Bibliography	591	(40)
Hints and Answers	593	(38)
Index	631

Excerpts

The introductory differential equations course plays an interesting role in the undergraduate mathematics curriculum. It is a required course for most science and engineering students, many of whom will take major courses that require certain knowledge of and skills related to differential equations. The course also plays the role of an introductory "applied course" for mathematics students, where modeling is often a primary focus. Our point of view, which in no way diminishes the utility of the course's content, is that the differential equations course has two overarching functions: (1) It is where students are introduced to the central subject in all of applied mathematics, a subject whose questions have spawned most of the classical theory of analysis since the time of Newton and continue to create a rich field of mathematical activity today. (2) It is a place to reinforce and extend the student's (perhaps tenuous) understanding of calculus. It is the next step after calculus on the path that leads to real analysis. Whether the student will follow that path any farther makes little difference. Throughout this book, we attempt to emphasize that the primary goal when studying a differential equation is always to understand the behavior of its solutions. Writing down formulas for solutions is just one means to that end. Qualitative and numerical methods are equally, if not more, important. Moreover, all of these "tools" are manifestations of theory, and we endeavor to emphasize that fact at a level which is appropriate for the course. Times have changed. Amazingly fast computers are commonplace, and powerful software systems such as Mathematica, Maple, and MATLAB make it possible to solve highly complex problems and create wonderfully illuminating graphics. This computing power has enormous potential for enhancing the differential equations course-perhaps more so than in any other mathematics course. Indeed, the computer can facilitate the analysis of solutions algebraically, numerically, and graphically. Yet the computer cannot teach the conceptual and theoretical foundations that this book strives to convey-but neither can tedious algebraic manipulations done with paper and pencil. This book does not emphasize technology, and only occasionally do we mention it directly. However, we recommend the use of a computer to facilitate those computations that require more than a few routine algebraic steps. We are platform-neutral with respect to technology, and our belief is that the best way to address technology specifically is with separate companion manuals. Thus we have created A Mathematica Companion for Differential Equations and A Maple Companion for Differential Equations for those who use Mathematica or Maple in their courses, and we highly recommend their use. Organization and Content For the most part, this book covers the traditional topics in the introductory, differential equations course, while enhancing the usual algebraic approach with more geometric ideas and interpretations. Naturally reflecting the biases of the author (an inkling of which should be provided by the preceding commentary), certain standard topics are omitted or given less emphasis, while others are given increased emphasis. A few of the topics covered are rather novel for an introductory text. The following are some of our more significant deviations from the "standard course": Systems and numerical methods are introduced early. Series methods are deemphasized, particularly the method of Frobenius. The method of undetermined coefficients is banished to near oblivion by the use of "exponential shift" and complex solutions. Linear equations of order three or higher are relegated to an appendix. An entire chapter is devoted to nonlinear systems in applications. We personally believe that this is where the subject of differential equations really becomes exciting and that s