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# Form the differential equation representing the family of ellipses having foci on x - axis and center at the origin.

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