I'm trying to understand the method of characteristics to solve first-order PDEs. As an example in his course, my professor solve this PDE for $u(x,y)$: $x\frac{\partial u}{\partial x}-y\frac{\partial u}{\partial y}=R $
with $u(s,s)=f(s)$ given along the parametrized curve $\Gamma$ defined by $x(s)=s$ and $y(s)=s$.
To determine the characteristics he first writes: $\frac{dx}{x}=-\frac{dy}{y}$ And then immediately jumps to: $\int^x_s \frac{dx'}{x'}=-\int^y_x\frac{dy'}{y'}$
I guess he simplifies a lot the resolution of the simple ODE but I don't really understand how he does that and how he finds the characteristics that way.