Question

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

Options

• `x  (dy)/(dx) - 2x` = 0

• `x  (dy)/(dx) - 2y` = 0

• `(dy)/(dx) - 2y` = 0

• `(dy)/(dx) - x` = 0

Answer 1

`x  (dy)/(dx) - 2y` = 0

Explanation:

The equation of a parabola with the vertex along the positive y-axis is-

`x^2 = 4ay`  .......(i)

Where (a) is the parameter 

Differentiating w.r.t. x

`2x = 4ay^1`  .......(ii)

Dividing (ii) by (i)

`(2x)/x^2 = (4ay)/(4ay)` ⇒ `y^-1/y = 2/x`

∴ `xy^1 = 2y`

Required differential equation is `x - 2y` = 0.