Differential Equations

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  • Edition: 4th
  • Format: Hardcover
  • Copyright: 2011-03-01
  • Publisher: Cengage Learning
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Incorporating an innovative modeling approach, this book for a one-semester differential equations course emphasizes conceptual understanding to help users relate information taught in the classroom to real-world experiences. Certain models reappear throughout the book as running themes to synthesize different concepts from multiple angles, and a dynamical systems focus emphasizes predicting the long-term behavior of these recurring models. Users will discover how to identify and harness the mathematics they will use in their careers, and apply it effectively outside the classroom.

Table of Contents

First-Order Differential Equations
Modeling via Differential Equations
Analytic Technique: Separation of Variables
Qualitative Technique: Slope Fields
Numerical Technique: Euler's Method
Existence and Uniqueness of Solutions
Equilibria and the Phase Line
Linear Equations
Integrating Factors for Linear Equations
First-Order Systems
Modeling via Systems
The Geometry of Systems
Analytic Methods for Special Systems
Euler's Method for Systems
The Lorenz Equations
Linear Systems
Properties of Linear Systems and the Linearity Principle
Straight-Line Solutions
Phase Planes for Linear Systems with Real Eigenvalues
Complex Eigenvalues
Special Cases: Repeated and Zero Eigenvalues
Second-Order Linear Equations
The Trace-Determinant Plane
Linear Systems in Three Dimensions
Forcing and Resonance
Forced Harmonic Oscillators
Sinusoidal Forcing
Undamped Forcing and Resonance
Amplitude and Phase of the Steady State
The Tacoma Narrows Bridge
Nonlinear Systems
Equilibrium Point Analysis
Qualitative Analysis
Hamiltonian Systems
Dissipative Systems
Nonlinear Systems in Three Dimensions
Periodic Forcing of Nonlinear Systems and Chaos
Laplace Transforms
Laplace Transforms
Discontinuous Functions
Second-Order Equations
Delta Functions and Impulse Forcing
The Qualitative Theory of Laplace Transforms
Numerical Methods
Numerical Error in Euler's Method
Improving Euler's Method
The Runge-Kutta Method
The Effects of Finite Arithmetic
Discrete Dynamical Systems
The Discrete Logistic Equation
Fixed Points and Periodic Points
Chaos in the Lorenz System
Changing Variables
The Ultimate Guess
Complex Numbers and Euler's Formula
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