Page 177 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
Both G (x, ) and {p(x)G (x, )} are bounded in the region x , a , x b. ...(10)
x x x
If a < x < b then as x x , keeping the relation x < and as x x , x , keeping the relation
0 0 0 0
x < , G(x, ) tends to finite values G (x + 0, x ) and G(x 0, x ) respectively, and ...(11)
x 0 0 0 0
1
G (x + 0, x ) G (x 0, x ) = ...(12)
x 0 0 x 0 0 ( p x )
0
G(x, ) = G( , x) ...(13)

Example: On the basis of equation (8), we have

d 2
L = , y (0) y (1) 0
x dx 2
x = 0, x = 1
Now solutions of
L (y) = 0
x
2
d y
or 2 = 0 ...(14)
dx
Suppose that a Green’s function G(x, ) exists. Then since

L (G(x, )) = 0 for x ,
x
G(x, ) must be represented, by means of a fundamental system y (x), y (x) of the solutions of
1 2
L (y) = 0, as follows:
x
2
d y
The general solution of 2 = 0.
dx
So the solution of (14) is
y = c x + c ...(15)
1 2
Let the two solutions be y (x) and y (x). Thus
1 2
if y (0) = 0 then c = 0
1 2
so y (x) = x, ...(16)
1
y (1) = 0 = c 1 + c = 0
2 1 2
c = c = 1
1 2
y = (1 x), ...(17)
2
Thus

C = 1 x ( 1) 1 (1 x ) 1

G(x, ) = 1 (1 )x (x )

= (1 x) (x > ). ...(18)

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