Page 89 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
t s
1 x 1
= e 1 t 1 s dx
(1 t )(1 s ) 0

t s
1 1 x 1 1 t 1 s
= e
(1 t )(1 s ) t s
1 0
(1 t ) (1 s )

1 (1 t )(1 s )
= [ 1]
(1 t )(1 s ) (1 t )(1 s ) t (1 s ) s (1 t )

1 1 2 3 n
st
st
st
= (1 st ) 1 st ( ) ( ) ( ) ...(iii)
1 st
m
In which coefficient of s t n
is 0 if m n ...(iv)

and is 1 if m n
Hence

0 if m n
x
e L m ( ) ( )dx =
L
x
x
n
0 1 if m n
or
x
x
x
e L m ( ) ( )dx = mn (where , , 1, 2, 3, ) ...(v)
L
n
m
n
0
Self Assessment
x
x
x
L
12. Whether e L 2 ( ) ( )dx is equal to
3
0
(a) 1 (b) 5
(c) 1 (d) 0
13. Find out
L ( )
x
mn m
m 0
14. Prove that

x
e L 1 ( ) ( )dx 0
L
x
x
2
0

82 LOVELY PROFESSIONAL UNIVERSITY