Form the differential equation representing the family of parabolas having vertex at origin and axis along the positive direction of the $x$-axis.

(N/A) Let $P$ denote the family of the above-said parabolas and let $(a, 0)$ be the focus of a member of the given family,where $a$ is an arbitrary constant. Therefore,the equation of the family $P$ is
$y^{2} = 4ax$ ...........$(1)$
Differentiating both sides of equation $(1)$ with respect to $x$,we get
$2y \frac{dy}{dx} = 4a$ ............$(2)$
Substituting the value of $4a$ from equation $(2)$ in equation $(1)$,we get
$y^{2} = \left(2y \frac{dy}{dx}\right)(x)$
or $y^{2} - 2xy \frac{dy}{dx} = 0$
which is the differential equation of the given family of parabolas.