Page 279 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes The complete integral is
(z 2 cy 2 ) ( cx b ) 2

Example 3: Solve by Charpit s method:
2
q = xp + p .
Solution:
Charpit s auxiliary equations are

dp dq dz dx dy f
p
p 0 0 ( p x 2 ) q ( 1) (x 2 ) 1 0
p
i.e. q = c from second term.
2
px + p = c

c
x (x 2 4 )
p = .
2
c
x (x 2 4 )
dz = dx + c dy.
2
x 2 1 x 4c
c
c
z = . (x 2 4 ) log{x (x 2 4 )} cy b .
4 2 2 4
dp dy
Aliter. Also , i.e., p = ae y
p 1
2 2y
y
q = axe + a e
y
y
2 2y
dz = ae dx + axe dy + a e dy.
a 2
2y
y
z = axe + e + b.
2
Example 4: Solve by Charpit s method:
(p + q) (px + qy) 1 = 0.
Solution:
By Charpit s method, auxiliary equations are
dp dq
= ...
( p p q ) 0 (p q )q

dp dq p
or c
p q q
2
q (1 + c) (cx + y) 1 = 0
1
or q =
(1 c )(cx y )

272 LOVELY PROFESSIONAL UNIVERSITY