Differential Equations and Linear Algebra

by Farlow, Jerry; Hall, James E.; McDill, Jean Marie; West, Beverly H.

• ISBN13: 9780130862501
• ISBN10: 0130862509
• Edition: 2nd
• Format: Hardcover
• Copyright: 2007-01-01
• Publisher: Prentice Hall

Summary

For sophomore-level courses in Differential Equations and Linear Algebra. &Extensively rewritten throughout, the Second Edition of this&flexible text features a seamless integration of linear algebra into the discipline of differential equations. Abundant computer graphics, IDE interactive illustration software, and well-thought-out problem sets make it an excellent choice for either the combination DE/LA course or pure differential equations courses. The authors' consistent, reader-friendly presentation encourages students to think both quantitatively and qualitatively when approaching differential equations &and reinforces concepts using similar methods to solve various systems (algebraic, differential, and iterative).

Table of Contents

Preface	xi
To the Reader	xix
Prologue	xxiii
First-Order Differential Equations
Dynamical Systems: Modeling	1	(8)
Solutions and Direction Fields	9	(10)
Separation of Variables: Quantitative Analysis	19	(8)
Euler's Method: Numerical Analysis	27	(10)
Picard's Theorem: Theoretical Analysis	37	(10)
Linearity and Nonlinearity
Linear Equations: The Nature of Their Solutions	47	(9)
Solving the First-Order Linear Differential Equation	56	(10)
Growth and Decay Phenomena	66	(9)
Linear Models: Mixing and Cooling	75	(7)
Nonlinear Models, Logistic Equation	82	(12)
Systems of DEs: A First Look	94	(15)
Linear Algebra
Matrices: Sums and Products	109	(19)
Systems of Linear Equations	128	(15)
The Inverse of a Matrix	143	(11)
Determinants and Cramer's Rule	154	(11)
Vector Spaces and Subspaces	165	(10)
Basis and Dimension	175	(16)
Second-Order Linear Differential Equations
The Harmonic Oscillator	191	(13)
Real Characteristic Roots	204	(14)
Complex Characteristic Roots	218	(12)
Undetermined Coefficients	230	(7)
Forced Oscillations	237	(12)
Conservation and Conversion	249	(12)
Linear Transformations
Linear Transformations	261	(15)
Properties of Linear Transformations	276	(12)
Eigenvalues and Eigenvectors	288	(16)
Coordinates and Diagonalization	304	(15)
Linear Systems of Differential Equations
Theory of Linear DE Systems	319	(14)
Linear Systems with Real Eigenvalues	333	(10)
Linear Systems with Nonreal Eigenvalues	343	(9)
Decoupling a Linear DE System	352	(6)
Stability and Linear Classification	358	(9)
Nonlinear Systems of Differential Equations
Nonlinear Systems	367	(10)
Linearization	377	(11)
Numerical Solutions	388	(8)
Chaos, Strange Attractors and Period Doubling	396	(9)
Forced Equations and Systems
Linear Nonhomogeneous Problems	405	(11)
Variation of Parameters	416	(9)
Laplace Transform I	425	(14)
Laplace Transform II	439	(20)
Forced Oscillations	459	(10)
Discrete Dynamical Systems
Iterative Equations	469	(13)
Linear Iterative Systems	482	(12)
Nonlinear Iterative Equations: Chaos Again	494	(15)
Control Theory
Feedback Controls	509	(11)
Introduction to Optimal Control	520	(12)
Pontryagin Maximum Principle	532	(9)
CN Complex Numbers and Complex-Valued Functions	541	(8)
LT Linear Transformations	549	(8)
PF Partial Fractions	557	(6)
SS Spreadsheets for Systems	563	(4)
Bibliography	567	(4)
Answers to Selected Problems	571	(62)
Index	633

Excerpts

This text is a response to departments of mathematics (many at engineering colleges) that have asked for a combined course in differential equations and linear algebra. It differs from other combined texts in its effort to stress the modern qualitative approach to differential equations, and to merge the disciplines more effectively. Differential Equations In recent years, the emphasis in differential equations has moved away from the study of closed-form transient solutions to the qualitative analysis of steady-state solutions. Concepts such as equilibrium points and stability are becoming the focus of attention, replacing phrases such as integrating factor and reduction of order, and diminishing concentration on formulas. In the past, students of differential equations were generally left with the impression that all differential equations could be "solved," and if given enough time and effort, closed-form expressions involving polynomials, exponentials, trigonometric functions, and so on could always be found. For students to be left with this impression is a mathematical felony inasmuch as even simple-looking equations such as dy/dt=y 2 -tanddy/dt= cos(y-t) do not have closed-form solutions. But these equations do have solutions, which we can see graphically, in Figures 1 and 2. In the traditional differential equations course, students spent much of their time grinding out "cookbook" solutions to "cookbook" equations, and not gaining much intuition and real understanding for the solutions or the subject. Nowadays, with computers and software packages readily available for finding numerical solutions, plotting vector and directional fields, and carrying out physical simulations, the student can study differential equations on a more sophisticated level than former students, and ask questions not contemplated by students (or teachers) in the past. Key information is transmitted instantly by visual presentations, especially when students can watch solutions evolve. We use graphics heavily in the text and in the problem sets. Linear Algebra The visual approach is especially important in making the connections with linear algebra. Although differential equations have long been treated as one of the best applications of linear algebra, in traditional treatments students tended to miss key links. It's a delight to hear those who have taken those old courses gasp with sudden insight when theyseethe role of eigenvectors and eigenvalues in phase portraits. Throughout the text we stick to the main theme from linear algebra that the general solution of a linear system is the solution to the associated homogeneous equation plus any particular solution. Consequently, for the first-order linear differential equation we solve the homogeneous equation by separation of variables, and then find a particular solution by a first-order variation of parameters method. Of course, we solve the second-order linear equations and linear systems using the same strategy, giving a more systematic approach to solving linear differential equations, as well as showing how concepts in linear algebra play an important role in differential equations. Differences from Traditional Texts Although we have more pages explicitly devoted to differential equations than to linear algebra, we have tried to provide all the basics of both that either course syllabus would normally require. But merging two subjects into one (while at the same time enhancing the usual quantitative techniques with qualitative analysis of differential equations) requires streamlining and simplification. The result should serve students well in subsequent courses and applications. Some Techniques De-emphasized Many of the specialized techniques used to solve small classes of differential equations are no longer i