Page 447 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Sachin Kaushal, Lovely Professional University

Notes Unit 27: Volterra Integral Equations and Linear
Differential Equations

CONTENTS

Objectives
Introduction

27.1 Relation between Linear Differential Equations and Volterra Integral Equations
27.2 Conversion of Volterra Integral Equation of Second Kind into a Differential Equation
27.3 Summary

27.4 Keywords
27.5 Review Questions

27.6 Further Readings
Objectives

After studying this unit, you should be able to:
 Know that the existence and uniqueness of the solution of differential equations leads us
to the integral equations
 See the relation between the integral equations and the linear differential equations with
initial conditions.

 Understand that the solution of the integral equation also satisfies a certain differential
equation with boundary conditions.

Introduction

The connection between a differential equation and integral equation should be seen clearly.

This connection helps us to solve certain differential equations by converting it into an integral
equation and vice versa.

27.1 Relation between Linear Differential Equations and Volterra

Integral Equations

In the unit 24 we had seen that a differential equation of first order or second order under certain
conditions is converted into an integral equation. This idea can be further explained in details in
this unit. Let us consider an nth order linear differential equation as follows:
n
d y d n 1 y d n 2 y
x
x
a 1 ( ) a 2 ( ) ...... a y f ( ) ...(1)
x
n
dx n dx n 1 dx n 2
It is assumed that the unknown functions y(x), f(x), a (x) a (x),...a (x) are continuous and
1 2 n
differentiable on the interval (a, b). The function y(x) satisfies the following initial conditions:
(n 1) (n 1)
y
y (0) y , (0) y , y (0) y ...... (0) y ... ...(2)
y
0 0 0 0
440 LOVELY PROFESSIONAL UNIVERSITY