The equation of the curve satisfying  $y{\left(\frac{dy}{dx}\right)}^2+(x-y)\frac{dy}{dx}-x=0$ is  

$$\begin{gathered} y{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}-y\frac{dy}{dx}-x=0 \\ \frac{dy}{dx}(y\frac{dy}{dx}+x)-1(y\frac{dy}{dx}+x)=0 \\ (\frac{dy}{dx}-1)(y\frac{dy}{dx}+x)=0 \\ \frac{dy}{dx}=1\&ydy=-xdy \\ \begin{array}{l}dy=dx\\ y=x+C_1\\ straightline\end{array} \\ \begin{array}{l}\frac{y^2}{2}=\frac{-x^2}{2}+C_2\\ x^2+y^2=2C_2\\ circle\end{array} \end{gathered}$$