Fundamentals of Differential Equations

by Nagle, R. Kent; Saff, E. B.

• ISBN13: 9780805350562
• ISBN10: 080535056X
• Edition: 3rd
• Format: Hardcover
• Copyright: 1993-01-01
• Publisher: Addison-Wesley

Summary

This text is in a flexible one-semester text that spans a variety of topics in the basic theory as well as applications of differential equations.

Table of Contents

(Most chapters end with a Chapter Summary, Review Problems and Group Projects.)

Introduction

Background

Solutions and Initial Value Problems

Direction Fields

The Approximation Method of Euler

First Order Differential Equations

Introduction: Motion of a Falling Body

Separable Equations

Linear Equations

Exact Equations

Special Integrating Factors

Substitutions and Transformations

Mathematical Models and Numerical Methods Involving First Order Equations

Mathematical Modeling

Compartmental Analysis

Heating and Cooling of Buildings

Newtonian Mechanics

Electrical Circuits

Improved Euler's Method

Higher-Order Numerical Methods: Taylor and Runge-Kutta

Linear Second Order Equations

Introduction: The Mass-Spring Oscillator

Homogeneous Linear Equations

The General Solution

Auxiliary Equations with Complex Roots

Nonhomogeneous Equations: the Method of Undetermined Coefficients

The Superposition Principle and Undetermined Coefficients Revisited

Variation of Parameters

Qualitative Considerations for Variable-Coefficient and Nonlinear Equations

A Closer Look at Free Mechanical Vibrations

A Closer Look at Forced Mechanical Vibrations

Introduction to Systems and Phase Plane Analysis

Interconnected Fluid Tanks

Elimination Method for Systems with Constant Coefficients

Solving Systems and Higher-Order Equations Numerically

Introduction to the Phase Plane

Coupled Mass-Spring Systems

Electrical Systems

Dynamical Systems, Poincaré Maps, and Chaos

Theory of Higher-Order Linear Differential Equations

Basic Theory of Linear Differential Equations

Homogeneous Linear Equations with Constant Coefficients

Undetermined Coefficients and the Annihilator Method

Method of Variation of Parameters

Laplace Transforms

Introduction: A Mixing Problem

Definition of the Laplace Transform

Properties of the Laplace Transform

Inverse Laplace Transform

Solving Initial Value Problems

Transforms of Discontinuous and Periodic Functions

Convolution

Impulses and the Dirac Delta Function

Solving Linear Systems with Laplace Transforms

Series Solutions of Differential Equations

Introduction: The Taylor Polynomial Approximation

Power Series and Analytic Functions

Power Series Solutions to Linear Differential Equations

Equations with Analytic Coefficients

Cauchy-Euler (Equidimensional) Equations

Method of Frobenius

Finding a Second Linearly Independent Solution

Special Functions

Matrix Methods for Linear Systems

Introduction

Linear Algebraic Equations

Matrices and Vectors

Linear Systems in Normal Form

Homogeneous Linear Systems with Constant Coefficients

Complex Eigenvalues

Nonhomogeneous Linear Systems

The Matrix Exponential Function

Partial Differential Equations

Introduction: A Model for Heat Flow

Method of Separation of Variables

Fourier Series

Fourier Cosine and Sine Series

The Heat Equation

The Wave Equation

Laplace's Equation

Appendices

Newton's Method

Simpson's Rule

Cramer's Rule

Method of Least Squares

Runge-Kutta Precedure for n

Equations

Answers to Odd-Numbered Problems

Index

Table of Contents provided by Publisher. All Rights Reserved.