If V =x f(u) and u = y/x,

determine the value of n such that

x^2(∂^2V / ∂x^2) + nxy( ∂^2V / ∂x∂y) + y^2 ( ∂^2V / ∂y^2) = 0

Need help to check my working..

∂V/∂x = xf'(u) + f(u)

∂^2V/∂x^2 = xf"(u) + f'(u) +f'(u)

For u = y/x,

∂u/∂x = -y/x^2

∂^2u/∂x^2 = 2y/x^3

Sub into ∂V/∂x and ∂^2V/∂x^2,

I get ∂V/∂x= -y/x + y/x =0

∂^2V/∂x^2= x(2y/x^3) - (2y/x^2) =0

∂V/∂y = xf'(u)

∂^2V/∂y^2 = xf"(u)

∂u/∂y = 1/x

∂^2u/∂x^2 = 0

Sub into ∂V/∂y and ∂^2V/∂y^2,

∂V/∂y = xf'(u) = 1

∂^2V/∂y^2 = xf"(u) = 0

And,

∂^2V / ∂x∂y= (∂/ ∂x)(∂V/∂y)<----------(Do i need to find ∂^2u / ∂x∂y?)

= (∂/ ∂x)xf'(u)<-----------( i am using ∂u/∂y for this function)

= x(0) + 1/x = 1/x

Sub into given eqn,

x^2(0) + nxy(1/x) + y^2(0) = 0

ny = 0

n=0

Is this correct?