Let $F$ be the family of ellipses whose centre is the origin and major axis is the $y$-axis. Then the differential equation of family $F$ is

$x y \frac{d^2 y}{d x^2}+\frac{d y}{d x}\left(x \frac{d y}{d x}-y\right)=0$

Let the equation of the family F of required ellipses be

$A x^2+B y^2=1$           ......(i)

It is a two parameter family of curves.

Differentiating (i) with respect to $x$, we get

$A x+B y y_1=0$            .....(ii)

Differentiating (ii) with respect to $x$, we get

$A+B y_1{ }^2+B y y_2=0$        .......(iii)

Multiplying (iii) by $x$ and subtracting from (ii), we get

$B \left\{y y_1-x y_1^2-x y y_2\right\}=0$

$\Rightarrow x y y_2+y_1\left(x y_1-y\right)=0 \Rightarrow x y \frac{d^2 y}{d x^2}+\frac{d y}{d x}\left(x \frac{d y}{d x}-y\right)=0$

This is the required differential equation.