Page 450 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 27: Volterra Integral Equations and Linear Differential Equations

27.2 Conversion of Volterra Integral Equation of Second Notes
Kind into a Differential Equation

We have seen that a linear differential equation with initial conditions can be expressed into a
Volterra integral equation. In this section we can show that an integral equation can also be
converted into a linear differential equation. To see that we take up the following example.

Example: Convert the integral equation

x
u
y ( ) 3x 4 2 sin x (x ) u 2 3(x ) u 2 y ( )du ...(1)
x
0
into the linear differential equation.
Before attempting the problem we know that
t
t
d b ( ) b ( ) db da
b
u
t
u
t
a
t
t
u
t
y
y
t
u
K ( , ) ( )du K ( , ) ( )du K [ , ( )] K [ , ( )] ...(2)
dt a ( ) a ( ) t t dt dt
t
using equation (2), differentiate (1) with respect to x, we have
x
y
u
x
x
y ( ) 3 2cos x (x ) x 2 3(x ) x 2 y ( ) 2(x u ) 3 ( )du
0
x
y
y
x
x
or y ( ) 3 2cos x 2 ( ) 2(x u ) 3 ( )du ...(3)
u
0
Differentiating (3) again, we have
x
y
y ( ) 2sin x 2 ( ) 2(x x ) 3 ( ) (2) ( )du
x
u
x
y
y
x
0
x
u
x
y
x
x
y
or y ( ) 2sin x 2 ( ) 3 ( ) 2 y ( )du ...(4)
0
Differentiating equation (4) again, we have
x
x
x
x
y
y
y ( ) 2cos x 2y ( ) 3 ( ) 2 ( )
x
or y ( ) 2y ( ) 3 ( ) 2 ( ) 2cos x ...(5)
x
x
y
x
y
Self Assessment
2. Convert the integral equation
x
u
y ( ) 2x 2 3x 3cos x 2(x ) u 3 3 (x ) u 2 6 y ( )du
x
0
27.3 Summary
 We have taken up the case of nth order differential equation and have seen how an
integral equation can be established.
 There is a strong connection between the initial value differential equation and the Volterra
integral equation of the second type or of first type.
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