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#.

1 Answer
Jun 24, 2018

#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"#