Ordinary Differential Equations: A First Course, 1st Edition (Hardback) book cover

Ordinary Differential Equations

A First Course, 1st Edition

By D. Somasundaram

Narosa

291 pages

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Hardback: 9780849309885
pub: 2001-11-06
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Description

Though ordinary differential equations is taught as a core course to students in mathematics and applied mathematics, detailed coverage of the topics with sufficient examples is unique.

Written by a mathematics professor and intended as a textbook for third- and fourth-year undergraduates, the five chapters of this publication give a precise account of higher order differential equations, power series solutions, special functions, existence and uniqueness of solutions, and systems of linear equations.

Relevant motivation for different concepts in each chapter and discussion of theory and problems-without the omission of steps-sets Ordinary Differential Equations: A First Course apart from other texts on ODEs. Full of distinguishing examples and containing exercises at the end of each chapter, this lucid course book will promote self-study among students.

Table of Contents

HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

Introduction

Preliminaries

Initial Value Problems

Boundary Value Problems

Superposition Principle

The Wronskian and Its Properties

Linear Dependence of Solutions

Reduction of Order

Method of Variation of Parameters

The Method of Variation of Parameters for the Non-Homogeneous Equation of n-th order

A Formula for the Wronskian

Homogeneous Linear Differential Equations with constant Coefficients

n-th Order Homogeneous Differential Equations with Constant Coefficients

Examples I

Exercises I

POWER SERIES SOLUTIONS

Introduction

The Taylor Series Method

Second Order Equations with Ordinary Points

Second Order Linear Equations with Regular Singular Points

Two Exceptional Cases

Gauss Hypergeometric Series

The Point at Infinity as a Singular Point

Examples II

Exercises II

FUNCTIONS OF DIFFERENTIAL EQUATIONS

Introduction

Legendre Functions

Legendre Series Expansion

Some Properties of Legendre Polynomials

Hermite Polynomials

Properties of Laguerre Polynomials

Properties of Bessel Functions

Bessel Series Expansion

Examples III

Exercises III

EXISTENCE AND UNIQUENESS OF SOLUTIONS

Introduction

Lipschitz Condition and Gronwall inequality

Successive Approximations and Picard Theorem

Dependence of Solutions on the Initial Conditions

Dependence of Solutions on the Functions

Continuations of the Solutions

Non-Local Existence of Solutions

Examples IV

Exercises IV

SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS

Introduction

Systems of First Order Equations

Matrix Preliminaries

Representation of n-th Order Equations as a System

Existence and Uniqueness of Solutions of System of Equations

Wronskian of Vector Functions

The Fundamental Matrix and its Properties

Non-Homogeneous Linear Systems

Linear Systems with Constant Coefficients

Linear Systems with Periodic Coefficients

Existence and Uniqueness of Solutions of systems

Examples V

Exercises V

REFERENCES

SOLUTIONS TO EXERCISES

INDEX

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations