Question

How to prove ?

Let z=f(x,y) and `ze^z-x^2y=0`.
Prove that `x(delz)/(delx)-y(delz)/(dely)=1-1/(1+z),ifz!=1`.

Answer 1

`ze^z-x^2y=0 qquad star`

Differentials:

`dz \ e^z + z e^z \ dz -2 xy \ dx - x^2 dy =0`

`bb( e^z( 1 + z) \ dz -2 xy \ dx - x^2 dy =0 )`

For the partials:

• `dy = 0`

`z_x = (2 xy)/ ( e^z (1+ z))`

• `dx = 0`

`z_y = x^2/ ( e^z (1+ z))`

`:. x z_x - y z_y = (2 x^2y - x^2 y)/ ( e^z (1+ z)) = ( x^2y )/ ( e^z (1+ z))`

From `star, qquad = (z e^z)/( e^z (1+ z))`

`= (1 + z - 1 )/( (1+ z)) = 1- 1/(1+z)`

Can see the case for `z ne "-1"` but not for `z ne "1"`