Page 143 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Hence the condition (vii) is necessary and sufficient both. In the vector form the equation (i) can
be written as
 
. A dr = 0
where

 ˆ ˆ ˆ and
A = Pi Q j Rk

dr = dxi ˆ dy j ˆ dzk ˆ
 
.
The necessary and sufficient condition then becomes A Curve A 0 i.e.
P Q R

= 0
x y z
P Q R

Self Assessment

1. Show that the differential equation
3
xz dx z dy 2y dz 0
is integrable.
2. Show that the differential equation
yz (y z )dx zx (z x )dy xy (x y )dz 0

is integrable.

8.3 Methods for Solving the Differential Equations

Pdx Qdy Rdx = 0 ...(1)
The condition for integrability of the above equation is
Q R R P P Q
P Q R = 0 ...(2)
z y x z y x

If the differential equation (1) is exact differential then its integral is of the form

y
x
u ( , , ) = c, ...(3)
z
Now
u u u
du = dx dy dz 0 ...(4)
x y z
Giving us the conditions

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

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