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

(N/A) The equation of the family of ellipses having foci on the $y$-axis and the centre at the origin is given by:
$\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$ --- $(1)$
Differentiating equation $(1)$ with respect to $x$,we get:
$\frac{2x}{b^{2}}+\frac{2yy'}{a^{2}}=0$
$\Rightarrow \frac{x}{b^{2}}+\frac{yy'}{a^{2}}=0$ --- $(2)$
Dividing by $x$,we get $\frac{1}{b^{2}} = -\frac{yy'}{a^{2}x}$.
Differentiating equation $(2)$ with respect to $x$ again:
$\frac{1}{b^{2}}+\frac{y' \cdot y' + y \cdot y''}{a^{2}} = 0$
Substituting $\frac{1}{b^{2}} = -\frac{yy'}{a^{2}x}$ into the equation:
$-\frac{yy'}{a^{2}x} + \frac{(y')^{2} + yy''}{a^{2}} = 0$
Multiplying by $a^{2}x$:
$-yy' + x(y')^{2} + xyy'' = 0$
Thus,the required differential equation is:
$xyy'' + x(y')^{2} - yy' = 0$