first order PDE question - stuck in last parts of finding the solution
I'm trying to solve the following pde,
u(x,y) u_x + u_y =0 with u(x,0) = p(x) for some known p(x)
where u_x defines the partial derivative of u(x,y) wrt x
after finding the characteristic curves and the first integrals i get the general solution is
F(x^2 - zy^2, z) = 0
At this point i'm not sure what to do next,
usually you can rewrite this as
f(x^2-zy^2) + z =0, however as z is contained in the argument you cant solve this for z without knowing what f is.
One thing i was thinking was just to make up some F and continue on, but that does not seem correct,
can anyone push me in the right direction?