Step 1:

Family of ellipses with foci on $x$-axis and centre at the origin $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$

Differentiating with respect to $x$ we get,

$\large\frac{2x}{a^2}+\frac{2y}{b^2}\frac{dy}{dx}$$=0$

$\Rightarrow \large\frac{y}{x}(\frac{dy}{dx})=\frac{-b^2}{a^2}$

Step 2:

Differentiating with respect to $x$ we get,

$\large\frac{y}{x}(\frac{d^2y}{dx^2})+\frac{xdy/dx-y}{x^2}\frac{dy}{dx}$$=0$

$xy\large\frac{d^2y}{dx^2}$$+x(\large\frac{dy}{dx})^2$$-y\large\frac{dy}{dx}$$=0$

Which is required differential equation.