First order ordinary differential equations involving powers of the slope 
Are there any general approaches to differential equations like
$$x-x\ y(x)+y'(x)\ (y'(x)+x\ y(x))=0,$$
or that equation specifically?

The problem seems to be the term $y'(x)^2$. Solving the equation for $y'(x)$ like a qudratic equation gives some expression $y'(x)=F(y(x),x)$, where $F$ is "not too bad" as it involves small polynomials in $x$ and $y$ and roots of such object. That might be a starting point for a numerical approach, but I'm actually more interested in theory now.
$y(x)=1$ is a stationary solution. Plugging in $y(x)\equiv 1+z(x)$ and taking a look at the new equation makes me think functions of the form $\exp{(a\ x^n)}$ might be involved, but that's only speculation. I see no symmetry whatsoever and dimensional analysis fails.

Hint:
$x-xy+y'(y'+xy)=0$
$(y')^2+xyy'+x-xy=0$
$(y')^2=x(y-1-yy')$
$x=\dfrac{(y')^2}{y-1-yy'}$
Follow the method in http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=234:
Let $t=y'$ ,
Then $\left(1+\dfrac{t^3(t-1)}{y^2(t-1)+1}\right)\dfrac{dy}{dt}=\dfrac{2t^2}{y(1-t)-1}+\dfrac{yt^3}{y(1-t^2)-1}$
