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

Step-by-step solution

Family of parabolas with vertex at origin and axis along positive Y-axis: x^2 = 4ay where a is an arbitrary constant. Differentiating both sides with respect to x yields: 2x = 4a dy dx Eliminate a by substituting 4a = x^2 y (for y 0 ): 2x = ( x^2 y ) dy dx Multiplying both sides by y and dividing by x (for x 0 ): 2y = x dy dx Rewriting gives the differential equation: x dy dx - 2y = 0 This corresponds to option A .