Differential Equations (with CD-ROM)

by Blanchard, Paul; Devaney, Robert L.; Hall, Glen R.

• ISBN13: 9780495012658
• ISBN10: 0495012653
• Edition: 3rd
• Format: Hardcover
• Copyright: 2005-09-19
• Publisher: Brooks Cole

Summary

Incorporating a modeling approach throughout, this exciting text emphasizes concepts and shows that the study of differential equations is a beautiful application of the ideas and techniques of calculus to everyday life. By taking advantage of readily available technology, the authors eliminate most of the specialized techniques for deriving formulas for solutions found in traditional texts and replace them with topics that focus on the formulation of differential equations and the interpretations of their solutions. Students will generally attack a given equation from three different points of view to obtain an understanding of the solutions: qualitative, numeric, and analytic. Since many of the most important differential equations are nonlinear, students learn that numerical and qualitative techniques are more effective than analytic techniques in this setting. Overall, students discover how to identify and work effectively with the mathematics in everyday life, and they learn how to express the fundamental principles that govern many phenomena in the language of differential equations.

Table of Contents

First-Order Differential Equations	1	(150)
Modeling via Differential Equations	2	(18)
Analytic Technique: Separation of Variables	20	(16)
Qualitative Technique: Slope Fields	36	(17)
Numerical Technique: Euler's Method	53	(12)
Existence and Uniqueness of Solutions	65	(11)
Equilibria and the Phase Line	76	(20)
Bifurcations	96	(16)
Linear Equations	112	(14)
Integrating Factors for Linear Equations	126	(25)
Review Exercises for Chapter 1	138	(6)
Labs for Chapter 1	144	(7)
First-Order Systems	151	(82)
Modeling via Systems	152	(17)
The Geometry of Systems	169	(18)
Analytic Methods for Special Systems	187	(11)
Euler's Method for Systems	198	(15)
The Lorenz Equations	213	(20)
Review Exercises for Chapter 2	220	(4)
Labs for Chapter 2	224	(9)
Linear Systems	233	(148)
Properties of Linear Systems and the Linearity Principle	234	(24)
Straight-Line Solutions	258	(16)
Phase Planes for Linear Systems with Real Eigenvalues	274	(16)
Complex Eigenvalues	290	(19)
Special Cases: Repeated and Zero Eigenvalues	309	(15)
Second-Order Linear Equations	324	(17)
The Trace-Determinant Plane	341	(13)
Linear Systems in Three Dimensions	354	(27)
Review Exercises for Chapter 3	370	(5)
Labs for Chapter 3	375	(6)
Forcing and Resonance	381	(70)
Forced Harmonic Oscillators	382	(15)
Sinusoidal Forcing	397	(12)
Undamped Forcing and Resonance	409	(12)
Amplitude and Phase of the Steady State	421	(12)
The Tacoma Narrows Bridge	433	(18)
Review Exercises for Chapter 4	443	(3)
Labs for Chapter 4	446	(5)
Nonlinear Systems	451	(108)
Equilibrium Point Analysis	452	(19)
Qualitative Analysis	471	(13)
Hamiltonian Systems	484	(18)
Dissipative Systems	502	(22)
Nonlinear Systems in Three Dimensions	524	(8)
Periodic Forcing of Nonlinear Systems and Chaos	532	(27)
Review Exercises for Chapter 5	549	(3)
Labs for Chapter 5	552	(7)
Laplace Transforms	559	(68)
Laplace Transforms	560	(12)
Discontinuous Functions	572	(9)
Second-Order Equations	581	(14)
Delta Functions and Impulse Forcing	595	(8)
Convolutions	603	(9)
The Qualitative Theory of Laplace Transforms	612	(15)
Table of Laplace Transforms	620	(1)
Review Exercises for Chapter 6	621	(3)
Labs for Chapter 6	624	(3)
Numerical Methods	627	(42)
Numerical Error in Euler's Method	628	(13)
Improving Euler's Method	641	(8)
The Runge-Kutta Method	649	(11)
The Effects of Finite Arithmetic	660	(9)
Review Exercises for Chapter 7	664	(1)
Labs for Chapter 7	665	(4)
Discrete Dynamical Systems	669	(54)
The Discrete Logistic Equation	670	(13)
Fixed Points and Periodic Points	683	(9)
Bifurcations	692	(9)
Chaos	701	(8)
Chaos in the Lorenz System	709	(14)
Review Exercises for Chapter 8	715	(2)
Labs for Chapter 8	717	(6)
APPENDICES	723	(26)
A Changing Variables	724	(12)
B The Ultimate Guess	736	(8)
C Complex Numbers and Euler's Formula	744	(5)
Hints and Answers	749	(70)
Index	819