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

$x \frac{d y}{d x}-y=0$

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

$y=m x$           ......(i)

$\Rightarrow \frac{d y}{d x}=m$       ......(ii)

From (i) and (ii), we get $y=x \frac{d y}{d x}$, which is the required differential equation.