Question

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

Answer 1

The equation of the family of ellipses having centre at the origin and foci on the x-axis is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.........(1)\]
where a and b are the parameters.
As this equation contains two parameters, we shall get a second-order differential equation.
Differentiating (1) with respect to x, we get
\[\frac{2x}{a^2} + \frac{2y}{b^2}\frac{dy}{dx} = 0..........(2)\]

Differentiating (2) with respect to x, we get

\[\frac{2}{a^2} + \frac{2}{b^2}\left[ \left( \frac{dy}{dx} \right)^2 + y\frac{d^2 y}{d x^2} \right] = 0\]

\[ \Rightarrow \frac{2}{a^2} = - \frac{2}{b^2}\left[ \left( \frac{dy}{dx} \right)^2 + y\frac{d^2 y}{d x^2} \right]\]

\[ \Rightarrow \frac{b^2}{a^2} = - \left[ \left( \frac{dy}{dx} \right)^2 + y\left( \frac{d^2 y}{d x^2} \right) \right] .........(3)\]

Now, from (2), we get

\[\frac{x}{a^2} = - \frac{y}{b^2}\frac{dy}{dx}\]

\[ \Rightarrow \frac{b^2}{a^2} = - \frac{y}{x}\frac{dy}{dx} ..........(4)\]
From (3) and (4), we get

\[- \frac{y}{x}\frac{dy}{dx} = - \left[ \left( \frac{dy}{dx} \right)^2 + y\left( \frac{d^2 y}{d x^2} \right) \right]\]

\[ \Rightarrow \frac{y}{x}\frac{dy}{dx} = \left[ \left( \frac{dy}{dx} \right)^2 + y\left( \frac{d^2 y}{d x^2} \right) \right]\]

\[ \Rightarrow y\frac{dy}{dx} = x \left( \frac{dy}{dx} \right)^2 + xy\left( \frac{d^2 y}{d x^2} \right)\]

\[ \Rightarrow xy\frac{d^2 y}{d x^2} + x \left( \frac{dy}{dx} \right)^2 - y\frac{dy}{dx} = 0\]

It is the required differential equation.