Differential Equations

by Blanchard, Paul

• ISBN13: 9780495561989
• ISBN10: 0495561983
• Edition: 4th
• Format: Hardcover
• Copyright: 2011-03-01
• Publisher: Cengage Learning

Summary

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

Bifurcations

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

Convolutions

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

Bifurcations

Chaos

Chaos in the Lorenz System

Appendices

Changing Variables

The Ultimate Guess

Complex Numbers and Euler's Formula

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