Equation of the family of an ellipse having foci on the $y$-axis and centers at the origin can be represented by $\frac{\mathrm{x}^{2}}{\mathrm{~b}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{a}^{2}}=1….(1)$
Differentiating the above equation with respect to $x$ on both sides, we have,
$\begin{array}{l}
\frac{2 x}{b^{2}}+\frac{2 y}{a^{2}} \frac{d y}{d x}=0 \\
\frac{x}{b^{2}}+\frac{y}{a^{2}} \frac{d y}{d x}=0 \\
\frac{y}{a^{2}} \frac{d y}{d x}=-\frac{x}{b^{2}} \\
\frac{y}{x} \frac{d y}{d x}=-\frac{a^{2}}{b^{2}}
\end{array}$
Again differentiating the above equation with respect to $x$ on both sides, we have,
$\begin{array}{l}
\frac{y}{x} \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\left(\frac{d y}{d x} x-y \frac{d x}{d x}\right)=0 \\
x y \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\left(\frac{d y}{d x} x-y \frac{d x}{d x}\right)=0
\end{array}$

Rearranging the above equation
$x y \frac{d^{2} y}{d x^{2}}+x\left(\frac{d y}{d x}\right)^{2}-y \frac{d y}{d x}=0$
This is the required differential equation.