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

.: Equation of ellipse having foci on $y$-axis and centre at origin is

$\cfrac{{{x^2}}}{{{b^2}}} + \cfrac{{{y^2}}}{{{a^2}}} = 1$

…(1)
Here, $a$ and $b$ are semi-major and semi-minor axes respectively.

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

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

$\Rightarrow \cfrac{{y{y_1}}}{x} = - \cfrac{{{a^2}}}{{{b^2}}}$
…(2)
Again differentiating (2) w.r.t. $x$,

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

$\Rightarrow x(y_1^2 + y{y_2}) - y{y_1} = 0$,

which is the required differential equation.