How to find characteristic of the following equation
$\mathbf{e^{-x}u_{xx}+2 \hspace{1pt} e^yu_{xy}+e^x \hspace{1pt} u_{x}=0}$
would the characteristic derived from the below equation?
$e^{-x}(\frac{\partial x}{\partial x})^2 - e^y \frac{\partial y}{\partial s} \frac{\partial x}{\partial s} + 0 = 0$
I am beginner to PDE, I know that the answer to this problem should be
$y=c\ \ \ or \ \ \ y=-ln(-2e^{x}+c)$,
I don't know how to get this solution, please give some hints.