$$ \begin{equation} \ \begin{cases} xu_x-yu_y=u, & U=\left\{\left(x,y\right):\:x>0,\:y>1\right\} \\ ^u|_{\Gamma }\:=\:g, & \Gamma =\left\{\left(x,y\right):\:x>0,\:y=1\right\} \end{cases} \end{equation} $$
This is the full PDE with all conditions
We just went through the theory of the Method of Characteristics (we basically did chapter 3.2 in Evan's book), but right now, I don't see a way how to start calculating this as we don't have a parametrization for our curve $\Gamma$ given (or do we?).
And because of that, I don't see a way to apply our "new knowledge" to that problem