Partial Differential Equation

For this exercise I will use the symbol of normal derivative for the partial derivatives:

Assume that f(x,y) is a differentiable function that obeys the partial differential equation

x*(df/dx)+y*(df/dy) = 0 (1)

in the region x>0, y>0. Suppose that the variables u and v are related to x and y by u=x/y and v=x*y

Use the chain rule to express the partial derivatives df/dx and df/dy in terms of derivatives with respect to u and v.

Hence write down the equation (1) in terms of u and v, and obtain the general solution of (1), expressing your final answer in terms of the original variables x and y.

My solution:

chain rule for f=f(u,v)

df/dx=df/du*du/dx + df/dv * dv/dx=df/du*(1/y) + df/dv * y

df/dy=df/du * du/dy + df/dv * dv/dy = df/du * (-x/y^2) + df/dv * x

Then (1) becomes (after some cancelling):

2*x*y*df/dv = 0, but x*y=v so we have 2v*df/dv=0, implies df/dv=0

=> f=g(u) for any function g(u)

=> f(x,y)=g(x/y)

I would appreciate if anyone can check my solution and let me know if it misses something! Thanks in advance for any help!