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

Explanation

The equation of the family of parabolas having vertex at the origin and axis along positive y-axis is


`x^{2}=4 ay ........(1)`, ‘a’ is a parameter.


Differentiating with respect to x,


`2 x=4 a frac{d y}{d x}`


`a=frac{x}{2 frac{d y}{d x}}`


Put the value of ‘a’ in equation 1, we get,


`x^{2}=4 times frac{x}{2(frac{d y}{d x})} times y`


`x frac{dy}{dx}=2 y`, is the differential equation.