Problem regarding unique solution of differential equation 
A unique  solution  to the  differential  equation  $y = x \frac{dy}{dx} - (\frac{dy}{dx})^2$ passing through $(x_0,y_0)$ doesnot exist

then choose the  correct option
$1.$ if $ x_0^2 > 4y_0$
$2.$ if $ x_0^2 = 4y_0$
$3.$ if $ x_0^2 < 4y_0$
$4.$ for any $(x_0 , y_0)$
My attempt  : Here $y = x \frac{dy}{dx} - (\frac{dy}{dx})^2$
Now i put  $x= e^z$ then $z= \log x$
So $y= Dy- D^2y$ 
$D^2y-Dy -y=0$
so $(D^2-D-1)y=0$
so auxiliary equation will  be $m^2-m-1=0$ ,$m= \frac{1 +_{-}\sqrt - 3}{2}$
so $y= e^{\frac{1}{2}x} (c_1 \cos(\frac{\sqrt - 3}{2} ) + c_2\sin ({\frac{\sqrt - 3}{2}} )x)$
After that im not able  to proceed further

Note that $y=bx-b^2$ is a solution, where
$$
b=\frac{-x_0\pm\sqrt{x_0^2-4y_0}}{2}
$$
This gives us two distinct solutions whenever $x_0^2\neq 4y_0$.  If instead $x_0^2=4y_0$, we can still find a second solution $y=x^2/4$.
