Question

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

Answer 1

The equation of the family of hyperbolas with the centre at origin and foci along the X-axis is
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1...\left(i\right)$
On differentiating both sides w.r.t. x, we get
$\frac{2x}{a^2}-\frac{2y{y}^{\prime }}{b^2}=0\Rightarrow \frac{y{y}^{\prime }}{x}=\frac{a^2}{b^2}$
Again differentiating w.r.t. x, we get
$\frac{x\frac{d}{dx}\left(y{y}^{\prime }\right)-y{y}^{\prime }.\frac{d}{dx}\left(x\right)}{x^2}=0$
(using quotient rule of differentiation)
$\Rightarrow x(y^{\prime }{)}^2+xy{y}^{\prime \prime }-y{y}^{\prime }=0\Rightarrow xy{y}^{\prime \prime }+x(y^{\prime }{)}^2-y{y}^{\prime }=0$
which is the required differential equation.