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

We know that the equation of the parabola having the vertex at origin and the axis along the positive y- axis is

x2 = 4ay ------(1)

Now, differentiating equation (1) w.r.t. x, we get,

2x = 4ay’ -----(2)

On dividing equation (2) by equation (1), we get,

xy’ = 2y

xy’ – 2y = 0

Therefore, the required differential equation is xy’ – 2y = 0.