$\displaystyle 2yu_x - xu_y = -4xyu^{\frac{1}{2}}$

Where $\displaystyle u_x$ and $\displaystyle u_y$ denotes partial differentiation with respect to x and y respectively of $\displaystyle u(x,y)$

For the above PDE how do I find the characteristic curves in the xy-plane?

How do I show that along these characteristics $\displaystyle u(x,y) = (y^2 +C)^2$ ? (C is a constant)

I think I found that $\displaystyle \frac{dx}{dt} = 2y$, and $\displaystyle \frac{dy}{dt} = -x$ and $\displaystyle \frac{du}{dt} = -4xyu^{\frac{1}{2}}$

and that on the characterisitcs $\displaystyle y^2 + \frac{x^2}{2} = C_1$ but I don't know where to go from here

Thanks