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