9780495012658

Differential Equations (with CD-ROM)

by ; ;
  • ISBN13:

    9780495012658

  • ISBN10:

    0495012653

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 9/19/2005
  • Publisher: Brooks Cole
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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

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