The differential equation of the family of lines through the origin, is

$x\frac{dy}{dx}-y=0$

The correct answer is option (1)  : $x\frac{dy}{dx}-y=0$

The equation of the family of lines passing through the origin is

$y = mx $ ............(i)

$⇒\frac{dy}{dx}= m$

From (i) and (ii), we get

$y=x\frac{dy}{dx}, $ which is the required differential equation.