Well, I'm learning this from a section in Gas Dynamics by Zucrow. It only presents one method...
It says to find the characteristic from \frac{dy}{dx} = \frac{b}{a}. In this case a = u, b = 1, c = -u.
So, \frac{dy}{dx} = \frac{1}{u}, integrating yields uy=x+c1 or y=\frac{x+c_{1}}{u}...