Page 160 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 8: Total Differential Equations, Simultaneous Equations

Therefore Notes
xdx ydy zdz = 0 ...(2)

x 2 y 2 z 2
or d = 0
2 2 2
or x 2 y 2 z 2 = constant = c ...(3)
1
Similarly

y dx xdy zdz y dx xdy zdz
yz (y x ) xz (x y ) z (x 2 y 2 ) = 0
Thus

ydx xdy zdz = 0

z 2
Thus xy = constant = c ...(4)
2 2
So the two integrals (3), (4) are complete integrals of (1) Q.E.D.

Example 2: Solve
dx dy dz
x 2 y 2 = 2xy (x y )z ...(1)

Solution: From the first two members
dx dy dz
x 2 y 2 2xy = (x y )z

or
dx dy dz
= ...(2)
x y z
Integrating (2) we have

log(x y ) = log z logc

x y = cz ...(3)
Also from (i)
dx dy dx dy
(x y 2 ) = (x y ) 2 ...(4)

Integrating (4) we have

(x y ) 1 = (x y ) 1 c 2 (c being a constant) ...(5)
2
1 1
or = c 2
x y x y

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