My question is related to another I asked some days ago about the method of characteristics. Although I now understand how the method works, I still don't understand how, for example given $x(s)$ and $y(s)$, we can solve $$\frac{dy}{-y}=\frac{dx}{x}$$ like that: $$\int^y_{y(s)}\frac{dy}{-y}=\int^x_{x(s)}\frac{dx}{x}$$
I don't understand why we use the parameter like that while integrating, could somebody show me more detailed steps ?