Question

Form the differential equation of all parabolas having the vertex at origin and axis along the positive $y$-axis.

Answer 1

Consider the all parabolas having the vertex at origin and axis along positive $y$-axis

$x^2=4ay$ ……… (1)

On differentiate with respect to $x$ ,we get

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

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

Put the value of a in equation (1), we get

$x^2=4\times \frac{x}{2\frac{dy}{dx}}\times y$

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

Hence, this is the answer.