Cauchy problem for PDE Cauchy problem $$y z_x-xz_y=0$$
and $ x_0(s)=cos (s),y_0(s)=sin(s),z_0(s)=1,s>0$ 
I use Lagrange's method 
$$\frac{dx}{y}=\frac{dy}{-x}=\frac{dz}{0}$$
From 1st and 2nd 
$x^2+y^2=c_1$ and from last relation $z=c_2$
So solution is of the type $z = f(x^2+y^2)$
By initial condition 
$$1=f(cos^2s+sin^2s) \implies 1 = f(1) $$  so  what i concluded from this about the solution to the problem ?  Please help .

Thankyou very much I got your point 
One more question : is there are condition  which quickly tell that Cauchy problem has unique solution, no solution, or infinitely many solution without solving  Cauchy problem ?
