Solving the following PDE? The PDE is given as 
$$y^2 u_x-xy u_y=x(u-2y)$$
What is the shortest method solving the PDE?

$y^2u_x-xyu_y=x(u-2y)$
$\dfrac{u_y}{y}-\dfrac{u_x}{x}=\dfrac{2}{y}-\dfrac{u}{y^2}$
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dy}{dt}=\dfrac{1}{y}$ , letting $y(0)=0$ , we have $\dfrac{y^2}{2}=t$
$\dfrac{dx}{dt}=-\dfrac{1}{x}$ , we have $-\dfrac{x^2}{2}=t+x_0=\dfrac{y^2}{2}+x_0$
$\dfrac{du}{dt}=\dfrac{2}{y}-\dfrac{u}{y^2}=\pm\dfrac{\sqrt2}{\sqrt t}-\dfrac{u}{2t}$ , we have $u(x,y)=\dfrac{F(x_0)}{\sqrt t}\pm\sqrt{2t}=\dfrac{f\left(x^2+y^2\right)}{y}\pm y$
