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.

The usual existence-and-uniqueness theory has problems  because $y'$ is not a Lipschitz function of $x$ and $y$.  In particular,  besides $y(x)=1$ the initial value $y(0)=1$ seems to have a series solution of the form
$$y = 1-\frac{1}{2}\,{x}^{2}+\frac{1}{6}\,{x}^{3}+{\frac {7}{48}}\,{x}^{4}+{\frac {1}{240}}
\,{x}^{5}+{\frac {1}{2160}}\,{x}^{6}+{\frac {787}{30240}}\,{x}^{7}+{
\frac {20047}{645120}}\,{x}^{8}+{\frac {370693}{10450944}}\,{x}^{9} + \ldots
 $$
