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

Let the equation of the family F of required ellipses be 

$A{x}^2+B{y}^2=1$         ... (i)

is a two parameter family of curves.

Differentiating (i) with respect to x, we get

$Ax+By{y}_1=0$        …(ii)

Differentiating (ii) with respect to x, we get

$A+B{y}_1^2+By{y}_2=0$        …(ii)

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

$B\left\{y{y}_1-x{y}_1^2-xy{y}_2\right\}=0$

$\Rightarrow xy{y}_2+y_1\left(x{y}_1-y\right)=0\Rightarrow xy\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}\left(x\frac{dy}{dx}-y\right)=0$

This is the required differential equation.