Differential Equations And Linear Algebra

Hardcover | December 26, 2006

byJerry Farlow, James E. Hall, Jean Marie Mcdill

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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).

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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 ...

Format:HardcoverDimensions:800 pages, 10.1 × 8 × 1.2 inPublished:December 26, 2006Publisher:Pearson EducationLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0131860615

ISBN - 13:9780131860612

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PREFACE Since the publication of the first edition of PSpice with Circuit Analysis, the changeover from the DOS to the Windows format and the proliferation of relatively cheap personal computers have created the need for this third edition of the text. The impact of such programs as PSpice will be profound in the workplace of the present and future electrical engineer. Also, they will increasingly affect the fashion in which electrical engineering and electronics technology are taught in colleges and technical schools. The analysis of electrical and electronic circuits even of modest size involves both complex and lengthy calculations. By means of PSpice, circuit complexity is far less a hindrance to a successful analysis of electrical circuit behavior. Relatively few rules of program syntax together with a few click-and-drag operations allow the electrical engineer and the student to solve complex circuits and produce circuit schematics of professional quality. The successful evaluation of a formula, when done by a calculator, gives the relationship between circuit variables at a particular operating point. PSpice, by contrast, allows for a global perspective of circuit behavior. The ease, compared to traditional methods, by which a frequency or transient (time) analysis can be performed well illustrates the point. Oscilloscope-like displays of circuit variables, their mathematical relationships, and concepts such as the RMS value of a signal can all be accurately and quickly displayed. PSpice allows a shift of emphasis away from computation of circuit variables toward their interpretations. It also allows a shift away from the analysis on the component level of circuits to the analysis of systems consisting of many circuits. Traditionally, students spend considerable time analyzing circuits containing a single bipolar transistor. However, practical circuits such as multistage amplifiers, operational amplifiers, active filters, and communication circuits all contain many transistors in addition to numerous other solid-state devices. Circuits of such complexity can be analyzed with relative ease by the PSpice program. Educators are challenged to incorporate these PSpice capabilities into the existing curriculum. With the pedagogical approach in this text, a particular circuit phenomenon is observed by means of graphical and numerical output data generated by PSpice. Having observed the phenomenon, the student is prompted to seek an explanation. At this point, mathematical concepts and formulas are introduced. The student is asked to show the correlation between data and calculation. A typical example of this approach is found in Chapter 6 of this text. There, the response of a RC circuit to a sinusoidal voltage is investigated by means of the PSpice program. Subsequent to that, the phasor method is introduced to solve for the circuit response. Finally, the results of that method and the PSpice data are compared. The structure of this book reflects this approach. Although its topical outline supports the traditional electrical engineering curriculum, a dual approach is taken. Every new circuit concept is introduced with its relevant PSpice commands. Subject matter and the PSpice program are used in a mutually supportive fashion. This text is neither an electrical engineering textbook with the PSpice program relegated to an appendix nor is it simply a reference manual for PSpice. The underlying assumption of this book-is that the combination of subject matter and the PSpice program can foster conceptual understanding and competence to advance the learning process. This text does not advocate the neglect of tradition teaching methods in favor of the use of the PSpice program; rather, it uses them in conjunction with the program. The material in this book is divided into ten chapters, which cover the topics usually found in a one-year course in electrical circuit analysis. Chapter 1 is devoted to the analysis of do circuits containing single and multiple independent current and voltage sources. The analysis of circuits containing dependent current and voltage sources is also included in this chapter. These devices play an important role in the modeling of many solid-state devices. Chapter 2 introduces some fundamental network theorems as applied to do circuits. The Superposition theorem, Thevenin's theorem, and Norton's theorems are demonstrated. Source conversions and the Maximum Power Transfer theorems conclude the chapter. Chapter 3 introduces the reader to the transient phenomena encountered in RC and RL circuits. The continuity of a capacitor's voltage and an inductor's current are related to the power flow and the energy content of these elements. The concept of the time constant is introduced. The circuit response to a linear pulse train is introduced. The model of a switch as used in PSpice is applied to circuits. Chapter 4 has as its subject the application of sinusoidal currents and voltage to resistive circuits. The latter are used so as not to introduce unnecessary complications at this time. The effects of time and phase shifts, their relation, and their effect on circuit behavior are studied. The sums of sinusoidal currents and voltages are obtained. Chapter 5 investigates the steady-state sinusoidal response of series, parallel RC, RL, and RLC circuits. The concept of impedance is demonstrated graphically by using PROBE plots. These plots demonstrate that admittance and impedance are steady-state concepts. The equivalency between various series and parallel circuits is demonstrated. Chapter 6 investigates the total response of series and parallel RC, RL, and RLC circuits. This response is shown to be the sum of the forced response due to the source and the transient, or natural, response of a circuit. The subject of electrical resonance investigated. Chapter 7 extends the concepts of Chapter 2 to circuits containing resistors, capacitors, and inductors. Superposition is applied to circuits containing both ac and do current and voltage sources. The ability of students to analyze such circuits will become increasingly important as they progress to electronic circuits containing solid state devices. It is demonstrated that Norton and Thevenin impedances are generally complex quantities. It is shown that maximum power from a source to a load will flow when the source impedance is the complex conjugate of the load impedance. Chapter 8 shows that power and energy flow in alternating current circuits. The power and energy relations for resistors, capacitors, and inductors are investigated. The concept of apparent, real, and reactive power is introduced and obtained by PSpice. It is important to note that this chapter provides the theoretical basis upon which the electrical utility industry is built. Chapter 9 introduces the reader to the frequency response of RC, RL, and RLC circuits. The necessary PSpice statements are introduced and applied. Both magnitude and phase plots are obtained and the impedance of circuits is plotted as a function of frequency. The concepts of the 3dB frequency are introduced and applied to the filters. The RC circuit is used both in its highpass and low-pass configurations. The effect of multiple sections on roll-off is demonstrated. The characteristics of a .band-pass filter are examined. The concept of the Q factor is covered. The relationship between transient analysis and AC (frequency) analysis is stressed. Chapter 10 applies non-sinusoidal current and voltage sources to electrical circuits. The ability of the PSpice program to perform a Fourier analysis is demonstrated. The Fourier transforms of some standard waves are obtained. The PSpice program calculates the total harmonic distortion (THD) of a wave. The RMS value of a wave train and the power it delivers to a load are calculated. The effects of half-wave and full-wave rectification on a Fourier spectrum are investigated. Wave symmetry and the effects of time shift of a wave are explored. Finally, a square pulse is applied to a RC high-pass filter and a band-pass filter. The effects of these circuits on the Fourier transform are investigated. Acknowledgments At the completion of this book, it is proper to thank those who contributed to it. Even a short reflective pause makes the author aware of his indebtedness to so many. It is impossible to tell how many years ago that the seed of this book was planted. There were the dedicated teachers at the City College of New York who introduced a young and untutored mind to the exciting world of electrical engineering. There are those by whose prior intellectual efforts this author has profited. By formal lecture and informal discussions over a cup of coffee, remarks were made and insights gained that found expression in this book. There are those with whom the author has had the privilege and pleasure of a congenial professional relationship. Among them is the late Professor Joseph Aidala, the former chairperson of the Electrical and Computer Engineering Technology Department at Queensborough Community College. Thanks are due to the present chairperson, Dr. Louis Nashelsky, whose professional competence is surpassed only by his human decency. Thanks is due to Professor Robert Boylestad, friend and colleague, for introducing me to various people at Prentice Hall who became interested in the publication of a book on PSpice. The work not only has to be written but it also needs to be published. This obliges me to thank all those involved in that part of the process. Initial thanks must go to Scott Sambucci, the Electronics Technology acquisitions editor at Prentice Hall. A special thanks goes to Rex Davidson, who guided the editing process through its various stages. Finally, a thank you to Linda Thompson, who did the editing and added to my humility. Writing on a more personal note, a special thanks must go to my dearest Daphine, who wanders with me along life's road. Thanks also to my two sons, Georg (without the final e) and Stefan, who, despite having been teenagers, turned into two wonderful adults.

Table of Contents

1 First-Order Differential Equations

1.1 Dynamical Systems: Modeling 1

1.2 Solutions and Direction Fields: Qualitative Analysis 11

1.3 Separation of Variables: Quantitative Analysis 25

1.4 Approximation Methods: Numerical Analysis 33

1.5 Picard’s Theorem: Theoretical Analysis 46


2 Linearity and Nonlinearity

2.1 Linear Equations: The Nature of Their Solutions 55

2.2 Solving the First-Order Linear Differential Equation 63

2.3 Growth and Decay Phenomena 73

2.4 Linear Models: Mixing and Cooling 80

2.5 Nonlinear Models: Logistic Equation 87

2.6 Systems of Differential Equations: A First Look 100


3 Linear Algebra

3.1 Matrices: Sums and Products 115

3.2 Systems of Linear Equations 130

3.3 The Inverse of a Matrix 146

3.4 Determinants and Cramer’s Rule 156

3.5 Vector Spaces and Subspaces 167

3.6 Basis and Dimension 177


4 Higher-Order Linear Differential Equations

4.1 The Harmonic Oscillator 195

4.2 Real Characteristic Roots 210

4.3 Complex Characteristic Roots 229

4.4 Undetermined Coefficients 244

4.5 Variation of Parameters 255

4.6 Forced Oscillations 261

4.7 Conservation and Conversion 274


5 Linear Transformations

5.1 Linear Transformations 285

5.2 Properties of Linear Transformations 300

5.3 Eigenvalues and Eigenvectors 311

5.4 Coordinates and Diagonalization 327


6 Linear Systems of Differential Equations

6.1 Theory of Linear DE Systems 343

6.2 Linear Systems with Real Eigenvalues 357

6.3 Linear Systems with Nonreal Eigenvalues 372

6.4 Stability and Linear Classification 384

6.5 Decoupling a Linear DE System 394

6.6 Matrix Exponential 400

6.7 Nonhomogeneous Linear Systems 410


7 Nonlinear Systems of Differential Equations

7.1 Nonlinear Systems 421

7.2 Linearization 431

7.3 Numerical Solutions 441

7.4 Chaos, Strange Attractors, and Period Doubling 449

7.5 Chaos in Forced Nonlinear Systems 456


8 Laplace Transforms

8.1 The Laplace Transform and Its Inverse 467

8.2 Solving DEs and IVPs with Laplace Transforms 475

8.3 The Step Function and the Delta Function 485

8.4 The Convolution Integral and the Transfer Function 499

8.5 Laplace Transform Solution of Linear Systems 509


9 Discrete Dynamical Systems

9.1 Iterative Equations 517

9.2 Linear Iterative Systems 530

9.3 Nonlinear Iterative Equations: Chaos Again 542


10 Control Theory

10.1 Feedback Controls 557

10.2 Introduction to Optimal Control 567

10.3 Pontryagin Maximum Principle 579