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(A) The equation of a parabola with vertex at the origin and axis along the positive $Y$-axis is given by $x^2 = 4ay$,where $a$ is an arbitrary consta

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$x \frac{dy}{dx} - 2y = 0$

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(A) The equation of a parabola with vertex at the origin and axis along the positive $Y$-axis is given by $x^2 = 4ay$,where $a$ is an arbitrary constant.To find the differential equation,we differentiate both sides with respect to $x$:$\frac{d}{dx}(x^2) = \frac{d}{dx}(4ay)$$2x = 4a \frac{dy}{dx}$$x = 2a \frac{dy}{dx}$From this,we get $2a = \frac{x}{dy/dx}$.Substitute $2a$ back into the original equation $x^2 = 2a(2y)$:$x^2 = \left( \frac{x}{dy/dx} \right) (2y)$$x^2 \frac{dy}{dx} = 2xy$$x \frac{dy}{dx} = 2y$$x \frac{dy}{dx} - 2y = 0$.Thus,the correct option is $A$.

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

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