The differential equation of all parabolas with axis paraIell to the axis of y, is

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$y_{2} = 2 y_{1} + x$

$y_{3} = 2 y_{1}$

$y_{2}^{3} = y_{1}$

none of these

answer is D.

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Detailed Solution

The equation of the family of parabolas with axis parallel to axis of y is

$(x - a)^{2} = A (y - b)$

This eguation contains three arbitrary constants viz. a, band A. Differentiating with respect to x, we get

$2 (x - a) = A \frac{d y}{d x}$

Differentiating again with respect to x, we get

$2 = A \frac{d^{2} y}{d x^{2}}$

Differentiating this with respect to $x$ we get $\frac{d^{3} y}{d x^{3}} = 0$

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