The differential equation of all ellipses centred at origin is :

$x y y_2+x y_1^2-y y_1=0$

Ellipse centred at origin are given by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$      ……(1)

where a and b are unknown constants

$\frac{2 x}{a^2}+\frac{2 y}{b^2} y_1=0 \Rightarrow \frac{x}{a^2}+\frac{y}{b^2} y_1=0$      ……(2)   

Differentiating again, we get

$\frac{1}{a^2}+\frac{1}{b^2}\left(y_1^2+yy_2\right)=0$      ……(3)

Multiplying (3) with x and then subtracting from (2) we get

$\frac{1}{b^2}\left(y y_1-x y_1^2-x y y_2\right)=0 \Rightarrow x y y_2+x y_1^2-y y_1=0$

Hence (2) is the correct answer.