Question

The differential equation of the family of hyperbolas having centres at the origin and whose axes are the co-ordinate axes is

• A. $xy{y}_2+{xy}_1^2+{yy}_1=0$
• B. $xy{y}_2-{xy}_1^2-{yy}_1=0$
• C. $xy{y}_2+{xy}_1^2-{yy}_1=0$
• D. $xy{y}_2+{xy}_1-{yy}_1=0$

Answer 1

The correct option is C $xy{y}_2+{xy}_1^2-{yy}_1=0$

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

$\frac{2x}{a^2}+\frac{2y}{b^2}\frac{dy}{dx}=0$

Implies that

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

Implies that

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

Implies that

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