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Home » Form the differential equation of the family of an ellipse having foci on the and centers at the origin.
Form the differential equation of the family of an ellipse having foci on the y-axis and centers at the origin.
CBSE | Class 12 | Differential Equations and Their Formation | Exercise 18C | Maths | RS Aggarwal
Form the differential equation of the family of an ellipse having foci on the y-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{\mathrm{x}^{2}}{\mathrm{~b}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{a}^{2}}=1....(1)
Differentiating the above equation with respect to x on both sides, we have,
\begin{array}{l} \frac{2 x}{b^{2}}+\frac{2 y}{a^{2}} \frac{d y}{d x}=0 \\ \frac{x}{b^{2}}+\frac{y}{a^{2}} \frac{d y}{d x}=0 \\ \frac{y}{a^{2}} \frac{d y}{d x}=-\frac{x}{b^{2}} \\ \frac{y}{x} \frac{d y}{d x}=-\frac{a^{2}}{b^{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{d y}{d x} x-y \frac{d x}{d x}\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.

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