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

(N/A) The standard equation of the family of hyperbolas with the centre at the origin and foci along the $x$-axis is:
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ --- $(1)$
Differentiating equation $(1)$ with respect to $x$,we get:
$\frac{2x}{a^{2}} - \frac{2yy'}{b^{2}} = 0$
$\Rightarrow \frac{x}{a^{2}} = \frac{yy'}{b^{2}}$ --- $(2)$
Differentiating again with respect to $x$ using the product rule:
$\frac{1}{a^{2}} = \frac{1}{b^{2}} (y' \cdot y' + y \cdot y'')$
$\Rightarrow \frac{1}{a^{2}} = \frac{1}{b^{2}} ((y')^{2} + yy'')$ --- $(3)$
Substitute the value of $\frac{1}{a^{2}}$ from equation $(3)$ into equation $(2)$:
$x \cdot \frac{1}{b^{2}} ((y')^{2} + yy'') = \frac{yy'}{b^{2}}$
Since $b^{2} \neq 0$,we multiply by $b^{2}$:
$x(y')^{2} + xyy'' = yy'$
Rearranging the terms,we get the required differential equation:
$xyy'' + x(y')^{2} - yy' = 0$