Page 174 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 10: Green’s Function Method

by multiplying equation (1) with Notes

x
x
x
exp p 1 ( )dx p ( ) ...(3)
a
and putting
q(x) = p (x)p (x) ...(4)
2
The coefficients p(x) and q(x) satisfy the following conditions:
p(x) and q(x) are real-valued continuous functions on the interval a x b and p(x) > 0 there.

dy
Putting z = p(x) in (2) we have
dx
dy z
= ...(5)
x
dx p ( )
dz
= q(x) y ...(6)
dx
If a pair of functions y(x) and z(x) is a solution of the equations (5) and (6) and if y(x) 0, then y(x),
and z(x) do not vanish at any point in the interval a x b. So due to y(x) 0, we may seek a
solution.
y(x) = (x) sin (x)

z(x) = (x) cos (x)
2
2
with p(x) = (y (x) + z (x)) 1/2 > 0 ...(7)
Substituting in (5) and (6) we have
d d ( )cos ( )
x
x
sin (x) + (x) cos (x) =
x
dx dx p ( )
d d
and cos (x) (x) sin (x) = q(x) (x) sin (x)
dx dx
Simplifying the above equations, we have

d ( ) 1
x
x
x
x
q ( ) P sin ( ) cos ( )
dx p ( ) ...(8)
x
x
d cos 2 ( ) 2
q ( )sin ( ), p ( ) 0
x
x
x
dx p ( )
x
The second equation of (8) does not contain the unknown , hence we can find a solution (x).
Then substituting this solution in the first equation, we can obtain the general solution p(x)
x
1
x
x
x
(x) = ( ) exp q ( ) sin ( )cos ( )dx ...(9)
x
p ( )
a
LOVELY PROFESSIONAL UNIVERSITY 167