Page 285 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Let ax + y = (a)u a dx + dy = (a) . du

dz a du du u 2 z 2 )
z (au 2 az 2 or dz z
)
This is homogeneous equation. To solve it put u = vz, then

dv 1
2 2
v z (v z z 2 )
dz z
dv 2
or z { (v 1) v }
dz
dz dv
or 2
z (v 1) v
dz 2
or { (v 1) v }dv
z

v 2 1 2 v 2
log z [(v 1)] log {v (v 1)} b
2 2 2

v 2 v 2 1 2
or log z (v 1) log{v (v 1)} b .
2 2 2
u ax y
This is a complete integral, where v
z z a

Example 12: Solve by Charpit s method:
2
2
2
2
(x y ) pq xy (p q ) 1 = 0. ... (1)
Solution:
2
2
2
2
f = (x y ) pq xy (p q ) 1 = 0
Charpit s auxiliary equations are
dp dq dx dy
2pqx z (p 2 q 2 ) 2ypq x (p 2 q 2 ) (x 2 y 2 )y 2pxy (x 2 y 2 )p 2pxy
from which it follows that each fraction

x dp y dq p dx q dy
=
0
(x dp + p dx) + (q dy + y dq) = 0
Integrating, px + qy = a
a qy
p = ... (2)
x

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