A solution of the differential equation $\left(\frac{d y}{d x}\right)^2-x \frac{d y}{d x}+y=0$ is :

y = 2x – 4

Let dy/dx = p 

$\Rightarrow p^2 -xp+ y = 0$

$y = xp - x^2$

$\frac{dy}{dx}=(x-2p)\frac{dp}{dx}+p$

$\Rightarrow p = (x-2p)\frac{dp}{dx}+p$

$\frac{dp}{dx} = 0$

$\Rightarrow p = \text{ constant}$

$\Rightarrow y = xc - c^2$

From options , c = 2 , y = 2x-4