Question

If U = `e^(xyz) f((xy)/z)` prove that `x(delu)/(delx)+z(delu)/(delx)2xyzu` and `y(delu)/(delx)+z(delu)/(delz)=2xyzu` and hence show that `x(del^2u)/(delzdelx)=y(del^2u)/(delzdely)`

Answer 1

U = `e^(xyz) f((xy)/z)`

`(delu)/(delx)=e^(xyz)[f'((xy)/z)xxy/z]+f((xy)/z)[e^(xyz)xxxy]`

`x(delu)/(delx)=e^(xyz)[(xy)/zf'((xy)/z)+xyz f((xy)/z)]`……………. (1)

`(delu)/(dely)=e^(xyz)[f'((xy)/z)xxx/z]+f((xy)/z)[e^(xyz) xx xz]`

`y(delu)/(dely)e^(xyz)[(xy)/zf'((xy)/z)+xyz f((xy)/z)]`……………….. (2)

`(delu)/(delz)=e^(xyz)[f'((xy)/z)xxy/z^2]+f((xy)/z)[e^(xyz)xxxy]`

`z(delu)/(delz)=e^(xyz)[-(xy)/zf'((xy)/z)+xyz f((xy)/z)]`……………….. (3)
Adding 1 and 3, we get

`x(delu)/(delx)+z(delu)/(delz)=2e^(xyz) xyz f((xy)/z)=2xyzu`
Adding 2 and 3, we get

`y(delu)/(dely)+z(delu)/(delz)=2e^(xyz) xyz f((xy)/z)=2xyzu`

For deduction,

`x(delu)/(delx)+z(delu)/(delz)=2xyzu`

Diff w.r.t z

`x(del^2u)/(delzdelx)+[z(del^2u)/(delz)+(delu)/(delz)(1)]=2xy[z(delu)/(delz)+u(1)]`

`x(del^2u)/(delzdelx)=(2xyz-)(delu)/(delz)-z(del^2u)/(delz^2)+2xyu`……………. (4)
Similarly,
`y(delu)/(dely)+z(delu)/(delz)=2xyzu`

Diff w.r.t z and solving, we get

`y(del^2u)/(delydelz)=(2xyz-1)(delu)/(delz)-z(del^2u)/(delz^2)+2xyu`…………….. (5)

∴ From 4 and 5, we get

`x(del^2u)/(delzdelx)=y(del^2u)/(delzdely)`