Page 209 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Richa Nandra, Lovely Professional University

Notes Unit 13: Orthogonality of Solutions

CONTENTS
Objectives
Introduction
13.1 Review of Some Basic Definitions

13.2 Review of Sturm Liouville Problem - Eigenvalues and Eigenfunctions
13.3 Review of Bessel’s Inequality and Completeness Relation
13.4 Orthogonality of Solutions of Some Equations
13.5 Summary
13.6 Keywords
13.7 Review Questions

13.8 Further Readings
Objectives

After studying this unit, you should be able to:
 Understand better the solutions of Bessel equations, Legendre equations, Hermite equations
and Laguerre differential equations.
 See that there are solutions which are obtained for some values of the parameters known
as eigenvalues. These solutions are known as eigenfunctions.
 Reduce these equations and many more differential equations of second order to Sturm-
Liouville boundary value problem. Hence the solutions can be shown to be orthogonal,
orthonormal and the set of various solutions of the equations form a complete set.

Introduction

Knowledge of Sturm Liouville problem and certain methods are prerequisite to the ideas of
orthogonality of the solutions of certain differential equations.

Also the solutions of these equations can be used to expand any function on an interval in terms
of them in a systematic manner.

13.1 Review of Some Basic Definitions

In the last four units we had studied the properties of linear second order differential equations.
By now you must have got enough inside into the solutions of the equations. It is seen that the
form of self-adjoint equations as well as Sturm Liouville’s boundary value problems led to the
kind of solutions of certain linear second order differential equations the orthogonal set of
functions which are solutions of these equations. The most important of these solutions are the
Fourier sine and cosine series, the Legendre polynomials, the Bessel functions; the Hermite
polynomials and Laguerre’s polynomials. In the last four chapters we had already seen that the
solutions do resemble the eigenfunctions of a self-adjoint operator and also form an orthogonal
set with respect to a weight factor. So it is advisable to introduce the inner product of two
functions. The concept of an orthogonal set of functions arises in a natural way from an analogy
with vectors in a vector space. This is a natural generalization of the concept of an orthogonal set

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