Page 471 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Another important fact is that to the given Kernel (1) and to the associated one
n
x
x
y
y
K ( , ) g k ( ) h k ( ) ...(21)
k 1
there corresponds the same function D( ) and consequently the same eigenvalues. This is because
the interchange of g and h carries a into a and hence only interchanges the rows and columns
k k hk kh
of determinant (8).
However, the eigenfunctions of the associated Kernel, i.e. the non-trivial solutions of the associated
homogeneous equation
x
y
( ) K ( , ) ( )dy 0 ...(22)
y
x
for = are not the previous function (16) but other ones,
0
n
x
x
r
( ) B * h ( ) (h 1, 2, ..., ), ...(23)
0h hk k
k 1
where
B * , B * , ... B *
11 12 1n
........................ ...(24)
B * , B * .... B *
r 1 r 2 rn
are any r linearly independent solutions of the associated homogeneous system

(1 a 11 ) 1 a 21 2 ... a n 1 n 0
a 21 1 (1 a 22 ) 2 ... a n 2 n 0,
 ...(25)
a 1n 1 a 2n 2 ... (1 a nn ) n 0

Any eigenfunction (x) corresponding to the eigenvalue and any associated eigenfunction
0h 0
(x) corresponding to a different eigenvalue are always orthogonal in the basic interval
1k 1
(0, 1).
In fact we have
x
x
y
x
y
I oh ( ) 1k ( )dx 0 1k ( )dx K ( , ) oh ( )dy
x
x
y
y
x
y
y
( )dy K ( , ) ( )dx 0 ( ) ( )dy 0 , I
0 0h 1k oh 1k
1 1
and this equality can be true only if = or if I = 0.
0 1
We now return to the non-homogeneous equation (1) for the case D( ) = 0. We prove that for
= the non-homogeneous equation can be solved if and only if the r orthogonality
0
conditions
( , oh ) f ( ) oh ( )dx 0 (h 1, 2, ..., ) ...(26)
r
x
f
x
are satisfied. In this case the non-homogeneous equation has r solutions of the form
x
( ) ( ) C ( ) C ( ) ... C ( ), ...(27)
x
x
x
x
1 01 2 02 r or
where (x) is a suitable linear combination of g (x), g (x),...g (x).
1 2 n
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