How to solve this First Order PDE $xu_x - yu_y + yu = y$? The PDE is $xu_x-yu_y+yu=y$ .
The method of characteristics gives $\dfrac{dx}{x}=-\dfrac{dy}{y}=\dfrac{du}{y-yu}$
Then $x=-c_1y$ and thus $c_1=-\dfrac{x}{y}$ .
Then, I did $du=\dfrac{(y-yu)dy}{y}$ to try to solve for the second constant so that I can have that in terms of $f\left(-\dfrac{x}{y}\right)$ but I can't seem to do that.

$xu_x-yu_y+yu=y$
$yu_y-xu_x=y(u-1)$
$u_y-\dfrac{x}{y}u_x=u-1$
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dy}{dt}=1$ , letting $y(1)=1$ , we have $y=t$
$\dfrac{dx}{dt}=-\dfrac{x}{y}=-\dfrac{x}{t}$ , letting $x(1)=x_0$ , we have $x=\dfrac{x_0}{t}=\dfrac{x_0}{y}$
$\dfrac{du}{dt}=u-1$ , we have $u(x,y)=f(x_0)e^t+1=f(xy)e^y+1$
