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

.: Equation of any hyperbola having foci on $x$-axis and centre at origin is

$\cfrac{{{x^2}}}{{{a^2}}} - \cfrac{{{y^2}}}{{{b^2}}} = 1$
…(1)

where, $a$ and $b$ are parameters.

Differentiating (1) w.r.t. $x$,

we get
$\cfrac{1}{{{a^2}}}2x - \cfrac{1}{{{b^2}}}2y{y_1} = 0 \Rightarrow \cfrac{{y{y_1}}}{x} = \cfrac{{{b^2}}}{{{a^2}}}$

…(2)
Again differentiating (2) w.r.t $x$,

we get
$\cfrac{{x\cfrac{d}{{dx}}\left( {y{y_1}} \right) - y{y_1}\left( 1 \right)}}{{{x^2}}} = 0 \Rightarrow x(y{y_2} + y_1^2) - y{y_1} = 0$

which is the required differential equation.