solution of variable coefficient equation Consider the equation $$u_x + yu_y = 0$$
and I know that this PDE has solution $u(x,y) = f(e^{-x}y)$
Can someone help me to derive this PDE to get the solution? Thank you

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$
$\dfrac{dy}{dt}=y$ , letting $y(0)=y_0$ , we have $y=y_0e^t=y_0e^x$
$\dfrac{du}{ds}=0$ , letting $u(0)=f(y_0)$ , we have $u(x,y)=f(y_0)=f(e^{-x}y)$
