Question

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

Answer 1

Step 1: Define the equation of hyperbola and differentiate it

The equation of hyperbola having foci on the x-axis and center at the origin is,

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ …(i)

Where, $a$ and $b$ are parameters.

Differentiating equation (i) with respect to $x$,

$\frac{2x}{a^2}-\frac{2y}{b^2}\frac{dy}{dx}=0$ $\mathrm{[}\mathrm{\because }\mathrm{}\frac{\mathrm{d}}{\mathrm{dx}}{\mathrm{x}}^{\mathrm{n}}\mathrm{=}{\mathrm{nx}}^{\mathrm{n}\mathrm{-}\mathrm{1}}\mathrm{,}\mathrm{}\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{C}\mathrm{=}\mathrm{0}\mathrm{]}$

$\Rightarrow$ $\frac{2x}{a^2}=\frac{2y}{b^2}\frac{dy}{dx}$

$\Rightarrow$ $\frac{x}{a^2}=\frac{y}{b^2}\frac{dy}{dx}$

$\Rightarrow$ $\frac{b^2}{a^2}x=y\frac{dy}{dx}$ …(ii)

$\Rightarrow$ $\frac{b^2}{a^2}=\frac{y}{x}\frac{dy}{dx}$ …(iii)

Differentiating equation (ii) with respect to $x$.

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

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

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

Step 2 : Compare the values of $\frac{b^2}{a^2}$

Using equation (iii) and equation (iv),

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

Or, $\frac{y}{x}{y}_1={\left(y_1\right)}^2+y\left(y_2\right)$ $\mathrm{[}\mathrm{\because }\mathrm{}\frac{\mathrm{dy}}{\mathrm{dx}}\mathrm{=}{\mathrm{y}}_{\mathrm{1}}\mathrm{,}\mathrm{}\frac{{\mathrm{d}}^{\mathrm{2}}\mathrm{y}}{{\mathrm{dx}}^{\mathrm{2}}}\mathrm{=}{\mathrm{y}}_{\mathrm{2}}\mathrm{]}$

$\Rightarrow$ $\frac{y}{x}{y}_1={y_1}^2+y{y}_2$

$\Rightarrow$ $y{y}_1=x\left[{y_1}^2+y{y}_2\right]$

$\Rightarrow$ $0=x\left({y_1}^2+y{y}_2\right)-y{y}_1$

Or, $x\left({y_1}^2+y{y}_2\right)-y{y}_1=0$

which is the required differential equation.

Hence, $x\left({y_1}^2+y{y}_2\right)-y{y}_1=0$ is the differential equation of the family of hyperbolas having foci on the x-axis and center at the origin