Page 483 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes in other words, for the general solution of (23 ).
r r r r
c j ( ) c j 1 j , c j 2 j , ..., c j jm
j
j 1 j 1 j 1 j 1
which contains a number r of arbitrary constants c , c , ..., c , there hold the following relations
1 2 r
m r
f c j j 0 ...(26 )
1 j 1
If the condition (26 ) is satisfied, then the general solution of (22 ) is given by the sum of a
r
particular solution ( 1 , 2 , ..., m ) of (22 ) and the general solution c j ( ) of (22 ), that
j
j 1
is, by the following expression containing r arbitrary constants c , c , ..., c .
1 2 r
r
c j ( ) ...(27)
j
j 1
r r r
c , c , ..., c
1 j 1 j 2 j 2 j m j jm
j 1 j 1 j 1
Similarly, the equations (23 ) admit a solution for given g , g , ...g if and only if the following
1 2 m
relations
m r
g c j j 0 ...(28)
1 j 1
hold; and under the condition (28), the general solution of (23 ) is given by the sum of a particular
r
j
solution ( 1 , 2 , ..., m ) of (23 ) and the general solution j 1 c p ( ) of (23 ), that is, by
j
the following expression containing r arbitrary constants c , c ,.... c .
1 2 r
r
j
c j ( )
j 1
r r r ...(29)
c , c , ..., c
1 j 1 j 2 j 2 j m j jm
j 1 j 1 j 1
Accordingly, substituting the solution- given by (27), if any, in (20), we obtain the general
solution (s) of the equation (1). The function (s) contains r arbitrary constants. In fact, if
m b r
r
r
0 ( ) ( , ) ( )dr c
s
s
v 1 v j jv
a
v 1 j 1
then, by the linear independence of (19)
r
0 c j jv (v = 1, 2, ..., m,)
j 1
This contradicts the fact that (1), (2), ... (r) are linearly independent solution of (22 ). We can
also obtain, substituting (29) in (24), the general solution (t) of (2) which contains a number r of
arbitrary constants.

476 LOVELY PROFESSIONAL UNIVERSITY