Question

The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of $ x-axis $ is

• A. $ y^2- 2xy \frac{dy}{dx} = 0 $
• B. $ y^2 + 2xy \frac{dy}{dx} = 0 $
• C. $ y^2- 2xy \frac{d^2y}{dx^2} = 0 $
• D. $ y^2 + 2xy \frac{d^2y}{dx^2} = 0 $

Answer 1

Answer: (A)

The equation of family of parabola having vertex at origin and axis along positive direction of axis of $x$ is given as
$y^{2}=4ax \dots(i)$
$\Rightarrow 2y \frac{dy}{dx}=4a$
$\Rightarrow 2y \frac{dy}{dx}$
$=\frac{y^{2}}{x}$ [using Eq. (i)]
$\Rightarrow \frac{y^{2}}{x}-2y \frac{dy}{dx}=0$
$\Rightarrow y^{2}-2xy \frac{dy}{dx}=0$