Step 1:

$\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$-----(i)

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

$\large\frac{2}{a^2}+\large\frac{2}{b^2}\bigg(\large\frac{dy}{dx}\bigg)^2+\frac{2y}{b^2}\frac{d^2 y}{dx^2}$$=0$-----(iii)

Step 2:

Eliminating $\large\frac{1}{a^2},\frac{1}{b^2}$ from (i),(ii),(iii)

$\begin{bmatrix}x^2 & y^2 & 1 \\ 2x & 2y \frac{dy}{dx} & 0 \\ 2 & 2 \bigg(\large\frac{dy}{dx}\bigg)^2 \normalsize +2y\frac{d^2y}{dx^2} & 0 \end{bmatrix}=0$

$4x \bigg[\bigg(\large\frac{dy}{dx}\bigg)^2 $$+y. \large\frac{d^2y}{dx^2}\bigg]-$$4y. \large\frac{dy}{dx}=0$

$xy \large\frac{d^2y}{dx^2}+x \bigg(\large\frac{dy}{dx}\bigg)^2$$-y \large\frac{dy}{dx}$$=0$ is the required D.E