Page 225 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Again differentiating with respect to y, we obtain
x 2
q = (1/z yq /z ) ...(11)
z 2
Eliminating from (10) and (11) we have

z xp ( yp )
=
( xq ) z yq
2
or z zxp zyq xypq = xypq
2
or z z (px + qy) = 0
or z = px + qy ...(12)

Example 3: Find the partial differential equation from the relation
2
2
2
x z 2 = (x y ) ...(13)
Solution: Differentiate (13) partially with respect to x keeping y fixed we have
z
2x 2z = 2x ...(14)
x
Again differentiate (13) partially with respect to y keeping x fixed.

z
2z = 2y ...(15)
y
Eliminating from (14) and (15) we have

2(x zp ) 2x
=
( 2zq ) ( 2 )
y
or xy zpy = xzq

or xzy + zpy = xy Ans ...(16)

Example 4: Find the partial differential equation from the relation

z = 1 (y 2 ) 2 (2y x ) ...(17)
x
Solution:

Differentiating (17) partially with respect to x keeping y fixed and z a dependent variable.
z
= 1 ( 2) 2 ( 1) ...(18)
x
Now differentiate (17) with respect to y,

z
= 2 ...(19)
y 1 2

Eliminating from (18) and (19) we have
2

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