Question

The differential equation by eliminating the arbitrary constants from $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

Answer 1

Given,

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

$\frac{x^2}{a^2}=1+\frac{y^2}{b^2}$

$\Rightarrow y^2=\left(\frac{b^2}{a^2}\right)x^2-b^2$

differentiating on both sides, we get,

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

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

$\therefore \left(\frac{b^2}{a^2}\right)=\frac{dy}{dx}\times \frac{y}{x}$

again differentiating on both sides, we get,

$\frac{d^{2}y}{d{x}^2}=\frac{1}{y}\left(\frac{b^2}{a^2}\right)$

$\frac{d^{2}y}{d{x}^2}=\frac{1}{y}(\frac{dy}{dx}\times \frac{y}{x})$

$\frac{d^{2}y}{d{x}^2}=\frac{1}{x}\frac{dy}{dx}$

$y^{\prime \prime }-\frac{1}{x}{y}^{\prime }=0$

$\therefore x{y}^{\prime \prime }-y^{\prime }=0$

is the reqired equation.