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Question Number 89178 by necxxx last updated on 15/Apr/20

If z(z^2 +3x)+3y=0 prove that   (∂^2 z/∂x^2 ) + (∂^2 z/∂y^2 )= ((2z(x−1))/((z^2 +x)^3 ))      please help.

Ifz(z2+3x)+3y=0provethat2zx2+2zy2=2z(x1)(z2+x)3pleasehelp.

Commented by niroj last updated on 16/Apr/20

   If  z(z^2 +3x)+3y=0 Prove that     (∂^2 z/∂x^2 ) +(∂^2 z/∂y^2 ) = ((2z(x−1))/((z^2 +x)^3 ))   D.w.r.to x      z^3 +3xz +3y=0      3z^2 .(dz/dx)+3z+3x(dz/dx)=0         3 (dz/dx)(z^2 +x)=−3z             (dz/dx)= ((−z)/(z^2 +x))=       (d^2 z/dx^2 )= ((−(dz/dx)(z^2 +x)+z(2z.(dz/dx)+1))/((z^2 +x)^2 ))      = ((−(dz/dx)(z^2 +x)+2z^2 (dz/dx)+z)/((z^2 +x)^2 ))     = (((dz/dx)(2z^2 −z^2 −x)+z)/((z^2 +x)^2 ))= ((((−z)/(z^2 +x))(z^2 −x)+z)/((z^2 +x)^2 ))   = ((xz−z^3 +z^3 +zx)/((z^2 +x)^3 ))= ((2zx)/((z^2 +x)^3 ))   Again,     D.w.r.to.y    z^3 +3xz+3y=0    3z^2 (dz/dy)+3x(dz/dy)+3=0       3(dz/dy)(z^2 +x)=−3       (dz/dy)= −1.(z^2 +x)^(−1)        (d^2 z/dy^2 )= −1.(−1)(z^2 +x)^(−2) .2z(dz/dy)             = ((2z)/((z^2 +x)^2 )).((−1)/((z^2 +x)))             = ((−2z)/((z^2 +x)^3 ))   now,    (∂^2 z/∂x^2 )+(∂^2 z/∂^2 y)= ((2xz)/((z^2 +x)^3 ))+((−2z)/((z^2 +x)^3 ))    =  ((2xz−2z)/((z^2 +x)^3 ))= ((2z(x−1))/((z^2 +x)^3 ))      hence   ∴   (∂^2 z/∂x^2 )+ (∂^2 z/∂y^2 )= ((2z(x−1))/((z^2 +x)^3 )) Proved//.

Ifz(z2+3x)+3y=0Provethat2zx2+2zy2=2z(x1)(z2+x)3D.w.r.toxz3+3xz+3y=03z2.dzdx+3z+3xdzdx=03dzdx(z2+x)=3zdzdx=zz2+x=d2zdx2=dzdx(z2+x)+z(2z.dzdx+1)(z2+x)2=dzdx(z2+x)+2z2dzdx+z(z2+x)2=dzdx(2z2z2x)+z(z2+x)2=zz2+x(z2x)+z(z2+x)2=xzz3+z3+zx(z2+x)3=2zx(z2+x)3Again,D.w.r.to.yz3+3xz+3y=03z2dzdy+3xdzdy+3=03dzdy(z2+x)=3dzdy=1.(z2+x)1d2zdy2=1.(1)(z2+x)2.2zdzdy=2z(z2+x)2.1(z2+x)=2z(z2+x)3now,2zx2+2z2y=2xz(z2+x)3+2z(z2+x)3=2xz2z(z2+x)3=2z(x1)(z2+x)3hence2zx2+2zy2=2z(x1)(z2+x)3Proved//.

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