I have the equation $$(y^2-u^2)u_x -xyu_y = xu$$ with the initial condition $u(x, x) = x$. I managed to write down the characteristic $$\frac{dx}{y^2-u^2} = \frac{dy}{-xy} = \frac{du}{xu}.$$ After some manipulations on the equation $$\frac{dy}{-xy} = \frac{du}{xu},$$ I got $$yu = c_1$$ for some constant $c_1$.
However, I couldn't manage to derive anything about $c_2$, the other constant which will show up in the solution.
Any help would be highly appreciated.
Hint: Can you think about multipliers?
$\frac{xdx+y{dy}}{xy^2-xu^2-xy^2} = \frac{du}{xu}$
$\implies xdx+ydy+udu=0$