Question

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

Answer 1

The equation of the family of hyperbolas having the centre at the origin and foci on the x-axis is \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1............(1)\]

where \[a\text{ and }b\]  are parameters.

As this equation contains two parameters, we shall get a second-order differential equation.

Differentiating equation (1) with respect to x, we get

\[\frac{2x}{a^2} - \frac{2y}{b^2}\frac{dy}{dx} =0..........(2)\]

Differentiating equation (2) with respect to x, we get

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

\[ \Rightarrow \frac{1}{a^2} = \frac{1}{b^2}\left[ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right]\]

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

Now, from equation (2), we get

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

\[ \Rightarrow \frac{b^2}{a^2} = \frac{y}{x}\frac{dy}{dx} ........(4)\]

From (3) and (4), we get

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

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

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

It is the required differential equation.