Form the differential equation of the family of hyperbolas having foci on the $x-axis$ and centers at the origin.

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Form the differential equation of the family of hyperbolas having foci on the $x-axis$ and centers at the origin.

Equation of the family of an ellipse having foci on the $y$ -axis and centers at the origin can be represented by $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $......(1)$
Differentiating the above equation with respect to $\mathrm{x}$ on both sides, we have,
$\begin{array}{l} \frac{2 x}{a^{2}}-\frac{2 y}{b^{2}} \frac{d y}{d x}=0 \\ \frac{x}{a^{2}}-\frac{y}{b^{2}} \frac{d y}{d x}=0 \\ \frac{y}{b^{2}} \frac{d y}{d x}=\frac{x}{a^{2}} \\ \frac{y}{x} \frac{d y}{d x}=\frac{b^{2}}{a^{2}} \end{array}$
Again differentiating the above equation with respect to $x$ on both sides, we have,
$\begin{array}{l} \frac{y}{x} \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\left(\frac{\frac{d y}{d x} x-y \frac{d x}{d x}}{x^{2}}\right)=0 \\ x y \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\left(\frac{d y}{d x} x-y \frac{d x}{d x}\right)=0 \end{array}$
Rearranging the above equation
$x y \frac{d^{2} y}{d x^{2}}+x\left(\frac{d y}{d x}\right)^{2}-y \frac{d y}{d x}=0$
This is the required differential equation.