Question

Obtain the differential equation of the family of parabolas having their focus at the origin and the axis along the x-axis ?

Answer 1

Hello,

Equation of parabola according to the question

${y}^{2} = 4ax$
differentiate with respect to x

$2y \frac{dy}{dx} = 4a \\ \\ y \frac{dy}{dx} = 2a$
put the value of a from eq1

$a = \frac{ {y}^{2} }{4x}$
$y \frac{dy}{dx} = 2 \times \frac{ {y}^{2} }{4x} \\ \\ \frac{dy}{dx} = \frac{y}{2x} \: \: \: \: \: ....eq2$
This is the required differential equation.

Hope it helps you