Solve the following Cauchy problem :
-xu_{x +}yu_{y = }1 for 0<x<y, u=2x on y=3x.
I managed to get the two constants C_{1 }and C_{2} as C_{1} = xy and C_{2} = lny - u.
Further to apply the initial conditions i found out the general solution for this equation as
u(x,y) = lny - f(xy).
Putting the initial conditions here ,i got f(3x^{2}) = ln(3x) - 2x.
What i cannot make out is , how do i find the value of f(xy) from the fact that f(3x^{2}) = ln(3x)-2x.
Need help urgently !