Page 155 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes The set of Auxiliary equations are
dx dy dz
= ...(2)
Q R R P P Q
z y x z y x

Q R
= 2y x , z x 2y
z y

R P
= 2x y , 2x y
x z
P Q
= , z z
y x

Q R
= 2y x x 2y
z y

R P
= 2x 2x ...(3)
x z
P Q
= z z 0
y x

Thus substituting (3) into equation (2) we have
dx dy dz
=
(2y x ) (x 2 ) (y 2 ) (2x y ) z ( z )
x
y
dx dy dz
= ...(4)
x
2(2y x ) 2(y 2 ) 0
Last equation gives dz = 0

or z = a = u (say) ...(5)
From first two members of equation (4) we have

dx dy
=
2y x y 2x
x
or (y 2 )dx = (2y x )dy

Re-arranging we have
ydx xdy 2xdx 2ydy = 0

or ( d xy d (x 2 ) d (y 2 ) = 0

( d xy x 2 y 2 ) = 0

Thus xy x 2 y 2 = constant = v (say) ...(6)

148 LOVELY PROFESSIONAL UNIVERSITY