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(B) The general equation of a parabola with its axis along the $y$-axis is $y = ax^2 + b$.Since there are two arbitrary constants $a$ and $b$, we diff

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

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(B) The general equation of a parabola with its axis along the $y$-axis is $y = ax^2 + b$.Since there are two arbitrary constants $a$ and $b$, we differentiate twice.Differentiating with respect to $x$: $\frac{dy}{dx} = 2ax$.Differentiating again with respect to $x$: $\frac{d^2y}{dx^2} = 2a$.From the first derivative, $a = \frac{1}{2x} \frac{dy}{dx}$.Substitute $a$ into the second derivative: $\frac{d^2y}{dx^2} = 2 \left( \frac{1}{2x} \frac{dy}{dx} \right) = \frac{1}{x} \frac{dy}{dx}$.Rearranging gives $x \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0$.

The differential equation of the family of all parabolas whose axis is the $y$-axis is ...

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