What is the differential equation of the family of parabolas having a vertex at origin and axis along positive y-axis?

Concept:

The equation of the parabola having a vertex at the (h,k) and axis along the positive y-axis is (x - h)² = 4a(y - k)

Calculation:

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

⇒ (x - 0)² = 4a(y - 0) 

⇒ x² = 4ay  __(i)

Differentiating both sides with respect to x,

⇒ 2x = 4a\(\frac{\text{dy}}{\text{dx}}\)

⇒ x = 2a \(\frac{\text{dy}}{\text{dx}}\) __(ii)

Putting the value of a from (i) in (ii),

⇒ x = 2\(x^2 \over 4y\) \(\frac{\text{dy}}{\text{dx}}\)

⇒ 2xy = x²\(\frac{\text{dy}}{\text{dx}}\)

⇒ x\(\frac{\text{dy}}{\text{dx}}\) − 2y = 0

∴ The correct option is (2).