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