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Fundamentals of Differential Equations with Boundary Value Problems

ISBN: 9780321145710 | 0321145712
Edition: 5th
Format: Hardcover w/CD
Publisher: Addison Wesley
Pub. Date: 1/1/2008

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SummaryTable of Contents
Fundamentals of Differential Equations, Sixth Edition is designed for a one-semester sophomore or junior-level course. Fundamentals of Differential Equations and Boundary Value Problems, Fourth Edition, contains enough material for a two-semester course that covers and builds on boundary-value problems. These tried-and-true texts help students understand the methods and concepts they will need to successfully complete engineering courses. The new texts retain the features that have made previous editions successful, while integrating recent advances in teaching and learning. The Fundamentals of Differential Equations and Boundary Value Problems version consists of the main text plus three additional chapters (Eigenvalue Problems and Sturm-Liouville Equations; Stability of Autonomous Systems; and Existence and Uniqueness Theory).
CHAPTER 1 INTRODUCTION1(36)
1.1 Background
1(5)
1.2 Solutions and Initial Value Problems
6(10)
... MORE
1.3 Direction Fields
16(8)
1.4 The Approximation Method of Euter
24(6)
Chapter Summary
30(1)
Technical Writing Exercises
30(1)
Group Projects for Chapter 1
31(6)
A. Taylor Series Method
31(1)
B. Picard's Method
32(1)
C. Magnetic Dipole
33(1)
D. The Phase Line
34(3)
CHAPTER 2 FIRST ORDER DIFFERENTIAL EQUATIONS37(50)
2.1 Introduction: Motion of a Falling Body
37(3)
2.2 Separable Equations
40(9)
2.3 Linear Equations
49(9)
2.4 Exact Equations
58(10)
2.5 Special Integrating Factors
68(4)
2.6 Substitutions and Transformations
72(8)
Chapter Summary
80(1)
Review Problems
81(1)
Technical Writing Exercises
82(1)
Group Projects for Chapter 2
83(4)
A. Torricelli's Law of Fluid Flow
83(1)
B. The Snowplow Problem
84(1)
C. Two Snowplows
84(1)
D. Clairaut Equations and Singular Solutions
85(1)
E. Asymptotic Behavior of Solutions to Linear Equations
86(1)
CHAPTER 3 MATHEMATICAL MODELS AND NUMERICAL METHODS INVOLVING FIRST ORDER EQUATIONS87(67)
3.1 Mathematical Modeling
87(2)
3.2 Compartmental Analysis
89(12)
3.3 Heating and Cooling of Buildings
101(7)
3.4 Newtonian Mechanics
108(11)
3.5 Electrical Circuits
119(4)
3.6 Improved Euler's Method
123(11)
3.7 Higher-Order Numerical Methods: Taylor and Runge-Kutta
134(10)
Group Projects for Chapter 3
144(10)
A. Aquaculture
144(1)
B. Curve of Pursuit
145(1)
C. Aircraft Guidance in a Crosswind
146(1)
D. Feedback and the Op Amp
147(1)
E. Bang-Bang Controls
148(1)
F. Price, Supply, and Demand
149(1)
G. Stability of Numerical Methods
150(1)
H. Period Doubling and Chaos
151(3)
CHAPTER 4 LINEAR SECOND-ORDER EQUATIONS
4.1 Introduction: The Mass-Spring Oscillator
154(6)
4.2 Homogeneous Linear Equations: The General Solution
160(9)
4.3 Auxiliary Equations with Complex Roots
169(10)
4.4 Nonhomgeneous Equations: The Method of Undetermined Coefficients
179(7)
4.5 The Superposition Principle and Undetermined Coefficients Revisited
186(8)
4.6 Variation of Parameters
194(4)
4.7 Qualitative Considerations for Variable-Coefficient and Nonlinear Equations
198(12)
4.8 A Closer Look at Free Mechanical Vibrations
210(10)
4.9 A Closer Look at Forced Mechanical Vibrations
220(8)
Chapter Summary
228(2)
Review Problems
230(1)
Technical Writing Exercises
231(1)
Group Projects for Chapter 4
232(9)
A. Undetermined Coefficients Using Complex Arithmetic
232(1)
B. An Alternative to the Method of Undetermined Coefficients
233(1)
C. Convolution Method
234(1)
D. Linearization of Nonlinear Problems
235(1)
E. Nonlinear Equations Solvable by First-Order Techniques
236(1)
F. Apollo Reentry
237(1)
G. Simple Pendulum
238(1)
H. Asymptotic Behavior of Solutions
239(2)
CHAPTER 5 INTRODUCTION TO SYSTEMS AND PHASE PLANE ANALYSIS241(76)
5.1 Interconnected Fluid Tanks
241(2)
5.2 Elimination Method for Systems with Constant Coefficients
243(10)
5.3 Solving Systems and Higher-Order Equations Numerically
253(11)
5.4 Introduction to the Phase Plane
264(15)
5.5 Coupled Mass-Spring Systems
279(7)
5.6 Electrical Systems
286(6)
5.7 Dynamical Systems, Poincare Maps, and Chaos
292(11)
Chapter Summary
303(1)
Review Problems
304(2)
Group Projects for Chapter 5
306(11)
A. The Growth of a Tumor
306(2)
B. Designing a Landing System for Interplanetary Travel
308(1)
C. Things That Bob
309(2)
D. Periodic Solutions to Volterra-Lotka Systems
311(1)
E. Hamiltonian Systems
312(2)
F. Strange Behavior of Competing Species-Part I
314(1)
G. Cleaning Up the Great Lakes
315(2)
CHAPTER 6 THEORY OF HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS317(31)
6.1 Basic Theory of Linear Differential Equations
317(9)
6.2 Homogeneous Linear Equations with Constant Coefficients
326(7)
6.3 Undetermined Coefficients and the Annihilator Method
333(5)
6.4 Method of Variation of Parameters
338(5)
Chapter Summary
343(1)
Review Problems
344(1)
Technical Writing Exercises
345(1)
Group Projects for Chapter 6
346(2)
A. Justifying the Method of Undetermined Coefficients
346(1)
B. Transverse Vibrations of a Beam
346(2)
CHAPTER 7 LAPLACE TRANSFORMS348(77)
7.1 Introduction: A Mixing Problem
348(3)
7.2 Definition of the Laplace Transform
351(9)
7.3 Properties of the Laplace Transform
360(6)
7.4 Inverse Laplace Transform
366(10)
7.5 Solving Initial Value Problems
376(8)
7.6 Transforms of Discontinuous and Periodic Functions
384(14)
7.7 Convolution
398(9)
7.8 Impulses and the Dirac Delta Function
407(7)
7.9 Solving Linear Systems with Laplace Transforms
414(3)
Chapter Summary
417(1)
Review Problems
418(1)
Technical Writing Exercises
419(2)
Group Projects for Chapter 7
421(4)
A. Duhamel's Formulas
421(1)
B. Frequency Response Modeling
422(2)
C. Determining System Parameters
424(1)
CHAPTER 8 SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS425(78)
8.1 Introduction: The Taylor Polynomial Approximation
425(6)
8.2 Power Series and Analytic Functions
431(9)
8.3 Power Series Solutions to Linear Differential Equations
440(11)
8.4 Equations with Analytic Coefficients
451(6)
8.5 Cauchy-Euler (Equidimensional) Equations
457(4)
8.6 Method of Frobenius
461(12)
8.7 Finding a Second Linearly Independent Solution
473(11)
8.8 Special Functions
484(12)
Chapter Summary
496(1)
Review Problems
497(1)
Technical Writing Exercises
498(1)
Group Projects for Chapter 8
499(4)
A. Spherically Symmetric Solutions to Schrodinger's Equation for the Hydrogen Atom
499(1)
B. Airy's Equation
500(1)
C. Buckling of a Tower
500(1)
D. Aging Spring and Bessel Functions
501(2)
CHAPTER 9 MATRIX METHODS FOR LINEAR SYSTEMS503(73)
9.1 Introduction
503(5)
9.2 Review 1: Linear Algebraic Equations
508(4)
9.3 Review 2: Matrices and Vectors
512(12)
9.4 Linear Systems in Normal Form
524(9)
9.5 Homogeneous Linear Systems with Constant Coefficients
533(12)
9.6 Complex Eigenvalues
545(6)
9.7 Nonhomogeneous Linear Systems
551(7)
9.8 The Matrix Exponential Function
558(9)
Chapter Summary
567(3)
Review Problems
570(1)
Technical Writing Exercises
571(1)
Group Projects for Chapter 9
572(4)
A. Uncoupling Normal Systems
572(1)
B. Matrix Laplace Transform Method
572(2)
C. Undamped Second-Order Systems
574(1)
D. Strange Behavior of Competing Species-Part II
575(1)
CHAPTER 10 PARTIAL DIFFERENTIAL EQUATIONS576(85)
10.1 Introduction: A Model for Heat Flow
576(3)
10.2 Method of Separation of Variables
579(10)
10.3 Fourier Series
589(18)
10.4 Fourier Cosine and Sine Series
607(5)
10.5 The Heat Equation
612(13)
10.6 The Wave Equation
625(13)
10.7 Laplace's Equation
638(13)
Chapter Summary
651(2)
Technical Writing Exercises
653(1)
Group Projects for Chapter 10
654(7)
A. Steady-State Temperature Distribution in a Circular Cylinder
654(1)
B. A Laplace Transform Solution of the Wave Equation
655(1)
C. Green's Function
656(2)
D. Numerical Method for Δu = f on a Rectangle
658(3)
CHAPTER 11 EIGENVALUE PROBLEMS AND STURM-LIOUVILLE EQUATIONS661(76)
11.1 Introduction: Heat Flow in a Nonuniform Wire
661(2)
11.2 Eigenvalues and Eigenfunctions
663(9)
11.3 Regular Sturm-Liouville Boundary Value Problems
672(12)
11.4 Nonhomogeneous Boundary Value Problems and the Fredholm Alternative
684(9)
11.5 Solution by Eigenfunction Expansion
693(6)
11.6 Green's Functions
699(9)
11.7 Singular Sturm-Liouville Boundary Value Problems
708(9)
11.8 Oscillation and Comparison Theory
717(9)
Chapter Summary
726(3)
Review Problems
729(1)
Technical Writing Exercises
730(1)
Group Projects for Chapter 11
731(6)
A. Hermite Polynomials and the Harmonic Oscillator
731(1)
B. Continuous and Mixed Spectra
731(1)
C. Picone Comparison Theorem
732(1)
D. Shooting Method
733(1)
E. Finite-Difference Method for Boundary Value Problems
734(3)
CHAPTER 12 STABILITY OF AUTONOMOUS SYSTEMS737(69)
12.1 Introduction: Competing Species
737(4)
12.2 Linear Systems in the Plane
741(13)
12.3 Almost Linear Systems
754(12)
12.4 Energy Methods
766(9)
12.5 Lyapunov's Direct Method
775(9)
12.6 Limit Cycles and Periodic Solutions
784(9)
12.7 Stability of Higher-Dimensional Systems
793(6)
Chapter Summary
799(2)
Review Problems
801(1)
Technical Writing Exercises
802(1)
Group Projects for Chapter 12
803(3)
A. Solitons and Korteweg-de Vries Equation
803(1)
B. Burger's Equation
803(1)
C. Computing Phase Plane Diagrams
804(1)
D. Ecosystem on Planet GLIA-2
805(1)
CHAPTER 13 EXISTENCE AND UNIQUENESS THEORY806
13.1 Introduction: Successive Approximations
806(7)
13.2 Picard's Existence and Uniqueness Theorem
813(8)
13.3 Existence of Solutions of Linear Equations
821(6)
13.4 Continuous Dependence of Solutions
827(7)
Chapter Summary
834(1)
Review Problems
835(1)
Technical Writing Exercises
835
APPENDICESA-1
A. Newton's Method
A-1
B. Simpson's Rule
A-3
C. Cramer's Rule
A-5
D. Method of Least Squares
A-6
E. Runge-Kutta Procedure for n Equations
A-9
ANSWERS TO ODD-NUMBERED PROBLEMSB-1
INDEXI-1

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