Page 152 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 8: Total Differential Equations, Simultaneous Equations

Notes
(1 v )du (1 u )dv dz
or = 0
(u v )(1 u ) z
[1 u u v ]du dv dz
= 0
(u v )(1 u ) u v z
1 1 dv dz
or du = 0
u v 1 u u v z

du dv du dz
= 0
u v 1 u z
Integrating we have

1 c being
log(u v ) log(1 u ) log z = log
c constant
or cz (u v ) = 1 u

or ( c x y )z = z + x ...(6)
is the solution of the equation (1).

Self Assessment

7. Solve the differential equation
2
z dx (z 2 2yz )dy (2y 2 yz zx )dz 0
8. Solve

(y 2 z 2 x 2 )dx 2xy dy 2xz dz 0

Method IV: Method of Auxiliary Equations

Let the given equation
P dx Q dy R dz = 0 ...(1)

be integrable. Then we must have

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

Comparing these two, we obtain

dx dy dz
=
dQ R R P P Q
dz y x z y x

These equations are called auxiliary equations and can be solved as shown in the two examples
below.

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