Page 199 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Richa Nandra, Lovely Professional University

Notes Unit 12: Sturm Comparison and Separation Theorems

CONTENTS
Objectives

Introduction
12.1 Linear Ordinary Second Order differential Equation

12.2 The Method of Reduction of Order
12.3 The Method of Variation of Parameters
12.4 The Wronskian

12.5 The Sturm Comparison Theorem
12.6 The Sturm Separation Theorem

12.7 Summary
12.8 Keywords
12.9 Review Questions

12.10 Further Readings

Objectives

After studying this unit, you should be able to:

 Deal with a linear second order differential equation with ease, there are a number of
important processes by which the solutions are found easily.
 Know that in certain important cases the method of reduction of order helps in solving the
differential equation.
 Discuss another method called the method of variation of parameters which helps in
solving non-homogeneous differentiation equation.

Introduction

Sturm comparison and separation theorems help us in understanding the nature of solutions of
certain differential equation where the solutions are periodic.
This process helps us in setting up the equation for Wronskian involving the solutions of the
differential equation.

12.1 Linear Ordinary Second Order Differential Equation

We here consider linear, second order ordinary differential equation of the form

2
d y dy
x
x
x
P ( ) 2 Q ( ) R ( )y F ( )
x
dx dx

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