Form the differential equation of the family of circles touching the $x$-axis at the origin.

(N/A) Let $C$ denote the family of circles touching the $x$-axis at the origin. Let $(0, a)$ be the coordinates of the center of any member of the family.
Therefore,the equation of the family $C$ is
$x^{2} + (y - a)^{2} = a^{2} \text{ or } x^{2} + y^{2} = 2ay$ ..........$(1)$
where $a$ is an arbitrary constant. Differentiating both sides of equation $(1)$ with respect to $x$,we get
$2x + 2y \frac{dy}{dx} = 2a \frac{dy}{dx}$
or $x + y \frac{dy}{dx} = a \frac{dy}{dx} \text{ or } a = \frac{x + y \frac{dy}{dx}}{\frac{dy}{dx}}$ ..........$(2)$
Substituting the value of $a$ from equation $(2)$ in equation $(1)$,we get
$x^{2} + y^{2} = 2y \left[ \frac{x + y \frac{dy}{dx}}{\frac{dy}{dx}} \right]$
or $\frac{dy}{dx}(x^{2} + y^{2}) = 2xy + 2y^{2} \frac{dy}{dx}$
or $\frac{dy}{dx}(x^{2} + y^{2} - 2y^{2}) = 2xy$
or $\frac{dy}{dx} = \frac{2xy}{x^{2} - y^{2}}$
This is the required differential equation of the given family of circles.