Page 153 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
Example 1: Solve

(y 2 yz z 2 )dx (z 2 zx x 2 )dy (x 2 xy y 2 )dz = 0 ...(1)

Here put P = y 2 yz z 2 , Q z 2 zx x 2

R = x 2 xy y 2 ...(2)

Q R
Now = 2z x , 2y x
z y

R P
= 2x y , 2z y
x z
P Q
= 2y z , 2x z
y x

The auxiliary equations are

dx dy dz
= ...(3)
Q R R P P Q
z y x z y x

dx dy dz
or = ...(4)
2(z y ) 2(x z ) 2(y x )
so

dx dy dz dx dy dz
= ...(5)
z y x z y x 0

Thus dx dy dz = 0
or x y z = constant = u (say) ...(6)

Also from (4)

(z y )dx = (x z )dy (y x )dz
z 2 y 2 x 2 z 2 y 2 x 2

(z y )dx (x z )dy (y x )dz
So ...(7)
0
Gives us

(z y )dx (x z )dy (y x )dz = 0 ...(8)

or ydx xdy zdy ydz zdx xdz = 0

or ( d xy yz zx ) = 0
So xy yz zx = constant = v (say) ...(9)

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