Form the differential equation representing the family of ellipses having foci on the $x$-axis and centre at the origin.

(N/A) The equation of the family of ellipses with foci on the $x$-axis and centre at the origin is given by:
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ............$(1)$
Differentiating equation $(1)$ with respect to $x$,we get:
$\frac{2x}{a^{2}} + \frac{2y}{b^{2}} \frac{dy}{dx} = 0$
$\frac{x}{a^{2}} + \frac{y}{b^{2}} \frac{dy}{dx} = 0$
$\frac{y}{b^{2}} \frac{dy}{dx} = -\frac{x}{a^{2}}$
$\frac{y}{x} \frac{dy}{dx} = -\frac{b^{2}}{a^{2}}$ ............$(2)$
Differentiating equation $(2)$ with respect to $x$ using the product rule:
$\frac{d}{dx} \left( \frac{y}{x} \frac{dy}{dx} \right) = \frac{d}{dx} \left( -\frac{b^{2}}{a^{2}} \right)$
$\left( \frac{y}{x} \right) \frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right) \frac{d}{dx} \left( \frac{y}{x} \right) = 0$
$\left( \frac{y}{x} \right) \frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right) \left( \frac{x \frac{dy}{dx} - y}{x^{2}} \right) = 0$
Multiplying by $x^{2}$:
$xy \frac{d^{2}y}{dx^{2}} + x \left( \frac{dy}{dx} \right)^{2} - y \frac{dy}{dx} = 0$
This is the required differential equation.