The differential equation of all ellipses with centres at the origin and the ends of one axis of symmetry at $( \pm 1,0)$, is

$\left(x^2-1\right) y'-x y=0$

Let the ends of other axis of symmetry be $(0, \pm a)$. Then, the equation of the ellipse is

$x^2+\frac{y^2}{a^2}=1 \Rightarrow a^2 x^2+y^2=a^2$             .....(i)

Differentiating w.r.to $x$, we get

$2 a^2 x+2 y \frac{d y}{d x}=0 \Rightarrow a^2=-\frac{y}{x} \frac{d y}{d x}$           .......(ii)

Eliminating $a^2$ from (i) and (ii), we get

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