Thread: Solving cauchy initial value problems for Partial differential equations

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 C₁ and C₂ as C₁ = xy and C₂ = 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(3x²) = ln(3x) - 2x.
What i cannot make out is , how do i find the value of f(xy) from the fact that f(3x²) = ln(3x)-2x.

Need help urgently !