Fundamentals of Differential Equations

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ISBN: 9780805350562 | 080535056X
Cover: Hardcover
Copyright: 1/1/1993

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Summary

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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.

(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, PoincarÃ© 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.

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Fundamentals of Differential Equations

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Table of Contents

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Fundamentals of Differential Equations

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Table of Contents

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