Page 469 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes is non-zero. So there is one and only one solution of the system of n simultaneous equations ,
1
, ... . Thus if D( ) 0, then the system (7) has one and only one solution given by Gamer’s rule
2 n
i.e.
1
n
nk n
1k
2k
k D b 1 D b 2 ... D b (k 1, 2, 3, ... )
D ( )
where D denotes co-factor of (h, k) the elements of the determinant (8), correspondingly, the
hk
solution (1) has the unique solution
n
y ( ) f ( ) k 1 D b 1 D b 2 ... D b g k ( ) ...(9)
x
x
x
2k
nk n
1k
D ( )
As D( ) 0, the corresponding Fredholm equation of the first kind
1
x
u
u
y ( ) K ( , ) ( )du ...(10)
y
x
0
has only the trivial solution y(x) 0 as D( ) 0.
31.2 Resolvent Kernel H(x, u, )
If we now substitute the expression of b in (5), the solution (9) can also be written as
i
1
u
x
y ( ) f ( ) D h 1 ( ) D h 2 ( ) D h 3 ( ) ... D h ( ) ( ) ( )du
u
g
u
x
u
f
x
u
2k
nk n
3k
k
1k
D ( ) 0
but the sum under the integral sign can be considered as the expansion of the negative of a
determinant of the (n + 1) order i.e.
u
u
g
(D h 1 ( ) D h 2 ( ) D h 3 (4) ...... D b ( ) ( )
u
x
ik
k
nk n
2k
3k
x
g
x
g
0 g ( ) ( ) ..................... ( )
x
1 2 n
h 1 ( ) 1 a 11 a 12 ......... a 1n
u
= ( , , )D x u  ...................................... a 2n ...(11)
.................................................
h ( ) (1 a )
u
n nn
Hence we can write equation (9) as
1
f
u
y ( ) f ( ) D ( , , ) ( )du ...(12)
x
x
u
x
D ( ) 0
Defining the resolvent Kernel H(x, u, ) by
u
D ( , , )
x
u
x
H ( , , ) ...(13)
D ( )
so equation (12) becomes
1
x
y ( ) f ( ) H ( , , ) ( )du ...(14)
x
u
u
f
x
0
In the equation (13) the resolvent Kernel H(x, u, ) is the quotient of two polynomials of the nth
degree in and the denominator is independent of x and u and this has important consequences.
462 LOVELY PROFESSIONAL UNIVERSITY