Solve $x=y\frac{dy}{dx}-\left(\frac{dy}{dx}\right)^{2}$

$x=y\dfrac{dy}{dx}-\left(\dfrac{dy}{dx}\right)^2$

$\left(\dfrac{dy}{dx}\right)^2-y\dfrac{dy}{dx}+x=0$

Apply the method in http://www.ae.illinois.edu/lndvl/Publications/2002_IJND.pdf#page=2:

Let $F(x,y,t)=t^2-yt+x~,$

Then $\dfrac{dy}{dt}=-\dfrac{t\dfrac{\partial F}{\partial t}}{\dfrac{\partial F}{\partial x}+t\dfrac{\partial F}{\partial y}}=-\dfrac{t(2t-y)}{1+t(-t)}=\dfrac{2t^2}{t^2-1}-\dfrac{ty}{t^2-1}$

$\dfrac{dy}{dt}+\dfrac{ty}{t^2-1}=\dfrac{2t^2}{t^2-1}$

$y=t+\dfrac{\ln(t+\sqrt{t^2-1})+C_1}{\sqrt{t^2-1}}$

$\therefore\dfrac{dx}{dt}=-\dfrac{\dfrac{\partial F}{\partial t}}{\dfrac{\partial F}{\partial x}+t\dfrac{\partial F}{\partial y}}=-\dfrac{2t-y}{1+t(-t)}=\dfrac{2t}{t^2-1}-\dfrac{y}{t^2-1}=\dfrac{t}{t^2-1}-\dfrac{\ln(t+\sqrt{t^2-1})+C_1}{(t^2-1)^\frac{3}{2}}$

$x=\int\biggl(\dfrac{t}{t^2-1}-\dfrac{\ln(t+\sqrt{t^2-1})+C_1}{(t^2-1)^\frac{3}{2}}\biggr)dt=\dfrac{t(\ln(t+\sqrt{t^2-1})+C_1)}{\sqrt{t^2-1}}+C_2$

Hence $\begin{cases}x=\dfrac{t(\ln(t+\sqrt{t^2-1})+C_1)}{\sqrt{t^2-1}}+C_2\\y=t+\dfrac{\ln(t+\sqrt{t^2-1})+C_1}{\sqrt{t^2-1}}\end{cases}$

Note that $$dx=\frac{dy}{p} \tag{*}$$ Since you have found $y$ in terms of $p$, you can find $dy$, substitute in $(*)$ and obtain the solution in a somewhat more palatable parametric form.