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dy
Sachin Kaushal, Lovely Professional University Unit 6: Existence Theorem for the Solution of the Equation dx = f(x, y)

Unit 6: Existence Theorem for the Notes
dy
Solution of the Equation = f(x, y)
dx

CONTENTS
Objectives
Introduction
6.1 On the solution of a Differential Equation
6.2 Picard’s Method
6.3 Remark on Approximate Solutions

6.4 Solutions by Power Series Expansion
6.5 Summary
6.6 Keyword
6.7 Review Questions
6.8 Further Readings

Objectives

After studying this unit, you will be able to:

 Discuss the existence and the uniqueness of the solution of the first order equation.
 Employ Picard’s method of finding the solution. The method consists in successive
approximation. It also leads to integral equations under certain conditions.

 Learn that the method is not so famous as it involves a lengthy set of solving integrals.

Introduction

The Picard’s method of finding the existence of the solution of first order equation is well
explained in Yosida’s book.

The method is quite general and can be applied to a system of n coupled first order differential
equations as well as equations of nth order. The case of nth order differential equation will be
taken up in the next unit.
6.1 On the Solution of a Differential Equation

In the previous units we have been studying different types of differential equations and their
solutions. Those differential equations chosen were for special purposes of studying certain
functions like Bessel function, Legendre polynomials, Hermite polynomials and Laguerre
polynomials. We also studied some differential equations which were easily soluble. In this
unit we want to study whether a given differential equation has a solution or not. We shall see
under what conditions the solution does exist.

An ordinary differential equation involves the dependent variable y, its derivatives
2
n
dy d y d y
, ..... , and independent variable x in the form of a functional relation
dx dx 2 dx n
LOVELY PROFESSIONAL UNIVERSITY 101