Page 448 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 27: Volterra Integral Equations and Linear Differential Equations

To convert the linear differential equation (1) into an integral equation we introduce a function Notes
(x) by the relation

n
d y ( )
x
dx n ...(3)
Integrating once we have by taking into account (2),
x
d n 1 y x
u
( )du
dx n 1 0
0
x
d n 1 y ( ) n 1 x
u
y 0 ( )du
dx n 1 0
Integrating once more we have
d n 2 y ( ) y n 2 y n 1 x x du u (u )du
x
dx n 2 0 0 0 0 1 1
x ...(4)
u
y n 0 2 y n 0 1 x (x u ) ( )du
0
In general integrating up to n times we have
x 2 x 3 2 x n 2 1 x n 1 1 x
y ( ) y 0 y x y 0 y 0 ... y n 0 y n 0 (x ) u n 1 ( )du ... ...(5)
u
x
0
2 3 n 2 (n 1)! (n 1)! 0
Writing (1) with the help of (3), (4) and (5) we have
 1 x   2 1 x 
u
x

x
u
 ( )  a 1 ( ) y n     ( )du  a 2 ( ) y n   xy n    (x  u ) ( )du 
x



0
0
0
 0   0 
x 2 1 x
a ( ) y n 3 xy n 2 y n 1 (x ) u 2 ( )du ...... ......
x
u
3 0 0 0
2 2 0
x 2 1 x n 1 1 x
a y y x y ...... y n (x ) n n 1 ( )du f ( )
x
u
n 0 0 0 0 ...(6)
2 (n 1)! (n 1)! 0
Defining
n 1 n 2 n 1 n 3 n 2 x 2 n 2
x
x
x
x
F ( ) f ( ) a 1 ( )y 0 a 2 ( ) y 0 xy 0 a y 0 xy 0 y 0
3
2
x 2 n 1 x n 1
...... a n y 0 xy 0 y 0 ... y 0 ...(7)
2 (n 1)!
and
x
a ( ) (x ) u n 1
u
x
K ( , ) a 1 ( ) a 2 ( ) (x u ) 3 (x u ) 2 ... a n
x
x
2! (x 1)!
n (x  ) u k  1
or K ( , )   a k ...(8)
x
u
k  1 (k  1)!
Substituting (7) and (8) into (6) we get
x
x
u
u
( ) K ( , ) ( )du F ( ) ...(9)
x
x
0
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