A Modern Introduction To Differential Equations

by Ricardo, Henry

• ISBN13: 9780618042395
• ISBN10: 0618042393
• Edition: 1st
• Format: Hardcover
• Copyright: 2002-01-15
• Publisher: Cengage Learning

Summary

Designed for a one-semester course,A Modern Introduction to Differential Equationspresents a solid and highly accessible introduction to differential equations, developing the concepts from a dynamic systems perspective and employing technology to treat topics graphically, numerically, and analytically. It thus blends the traditional differential equations text with the best of the more recent reform approaches. Written completely from the perspective of the student, the text provides strong pedagogy along with an informal, interactive style to help students develop an effective understanding of important techniques and applications. Worked out examples and exercises taken from biology, chemistry, economics, physics, and engineering enable students to see how math applies to their chosen majors. Coverage of a graphing calculator and/or CAS (computer algebra system) is fully integrated into the text to help students construct the required graphs and numerical approximations.

Table of Contents

Introduction to Differential Equations

Introduction

Basic Terminology

Solutions of Differential Equations

Technology and Differential Equations

Summary Project 1-1

First-Order Differential Equations

Introduction

Separable Equations

Linear Equations

Slope Fields

Phase Lines and Phase Portraits

Equilibrium Points: Sinks, Sources, and Nodes

Bifurcations

Existence and Uniqueness of Solutions

Summary Project 2-1 Project 2-2

The Numerical Approximation of Solutions

Introduction

Euler's Method

A Few More Words About Error

The Improved Euler Method

More Sophisticated Numerical Methods: Runge-Kutta and Others

Summary Project 3-1

Second- and Higher-Order Equations

Introduction

Homogeneous Second-Order Linear Equations with Constant Coefficients

Nonhomogeneous Second-Order Linear Equations with Constant Coefficients

Higher-Order Linear Equations with Constant Coefficients

Higher-Order Equations and Their Equivalent Systems

Qualitative Analysis of Autonomous Systems

Existence and Uniqueness

Numerical Solutions

Summary Project 4-1

Systems of Linear Differential Equations

Introduction

Systems and Matrices

Two-Dimensional Systems of First-Order Linear Equations

Stability of Linear Systems: Unequal Real Eigenvalues

Stability of Linear Systems: Equal Real Eigenvalues

Stability of Linear Systems: Complex Eigenvalues

Nonhomogeneous Systems

Generalizations: The n x n Case (n greater or equal to 3)

Summary Project 5-1

The Laplace Transform

Introduction

The Laplace Transform of Some Important Functions

The Inverse Transform and the Convolution

Transforms of Discontinuous Functions

Transforms of Impulse Functions--the Dirac Delta Function

Transforms and Systems of Linear Differential Equations

Qualitative Analysis via the Laplace Transform

Summary Project 6-1

Systems of Nonlinear Differential Equations

Introduction

Equilibria of Nonlinear Systems

Linear Approximation at Equilibrium Points

Two Important Examples of Nonlinear Equations and Systems

Van der Pol's Equation and Limit Cycles

Summary Project 7-1

Some Calculus Concepts and Results

Vectors and Matrices

Complex Numbers

Series Solutions of Differential Equations

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