Question

Form the differential equation representing the family of parabolas having a vertex at origin and axis along the positive direction of the $x$-axis .

• A. $\frac{dy}{dx}=\frac{x}{y}$
• B. $\frac{d^{2}y}{d{x}^2}=0$
• C. $\frac{dy}{dx}=\frac{y}{2x}$
• D. None of the above.

Answer 1

The correct option is C

$\frac{dy}{dx}=\frac{y}{2x}$

The equation for the parabola having a vertex at the origin and axis along the positive direction of x-axis be,

$y^2=4ax$------(1)

taking derivative on both the sides

$2y\frac{dy}{dx}=4a$

$\frac{dy}{dx}=\frac{4a}{2y}$

$\frac{dy}{dx}=\frac{2a}{y}$

from (1)

$\frac{dy}{dx}=\frac{y^2}{2xy}$

So, the differential equation for the parabola having a vertex at the origin and axis along the positive direction of x-axis be,

$\frac{dy}{dx}=\frac{y}{2x}$