I am trying to solve the equation $xU_x + yU_y = 0$. The characteristic equation is $\frac{dy}{dx} = \frac{y}{x}$. Hence $\frac{1}{y}dy = \frac{1}{x}dx,$ so $\ln y = \ln x + c_0$. This implies that $y = Cx$.
Now here, normally I would make a change of variables, but I am unsure what that change should be.
You already got the solution.
The general solution of this PDE is of the form $$u(x,y)=f(C)$$
Hence $$u(x,y)=f(\frac{y}{x})$$
You can determine the explicit (or implicit) expression if you have been give some auxiliary (initial) condition.