Question

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):

\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

 

Answer 1

The equation of family of curves is \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.............(1)\]

where `a` and `b` are parameters.

As this equation has two arbitrary constants, we shall get a differential equation of second order.

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

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

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

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

Now, from (2), we get

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

\[ \Rightarrow \frac{b^2}{a^2} = \frac{y}{x}\frac{dy}{dx} ..........\left( 4 \right)\]
From (3) and (4), we get 
\[\frac{y}{x}\frac{dy}{dx} = \left[ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right]\]
\[ \Rightarrow x\left[ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right] = y\frac{dy}{dx}\]
It is the required differential equation.