Fundamentals of Differential Equations with Boundary Value Problems

by Nagle, R. Kent; Saff, Edward B.; Snider, Arthur David

• ISBN13: 9780321145710
• ISBN10: 0321145712
• Edition: 5th
• Format: Hardcover w/CD
• Copyright: 2008-01-01
• Publisher: Addison Wesley

Summary

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).

Table of Contents

CHAPTER 1 INTRODUCTION	1	(36)
1.1 Background	1	(5)
1.2 Solutions and Initial Value Problems	6	(10)
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 EQUATIONS	37	(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 EQUATIONS	87	(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 ANALYSIS	241	(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 EQUATIONS	317	(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 TRANSFORMS	348	(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 EQUATIONS	425	(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 SYSTEMS	503	(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 EQUATIONS	576	(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 EQUATIONS	661	(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 SYSTEMS	737	(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 THEORY	806
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
APPENDICES	A-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 PROBLEMS	B-1
INDEX	I-1