The differential equation of the family of parabolas with focus at the origin and the x-axis as axis is

[EAMCET 2003]

$y {(\frac{d y}{d x})}^{2} + 4 x \frac{d y}{d x} = 4 y$

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$- y {(\frac{d y}{d x})}^{2} + 2 x \frac{d y}{d x} = - y$

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$y {(\frac{d y}{d x})}^{2} + y = 2 x y \frac{d y}{d x}$

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$y {(\frac{d y}{d x})}^{2} + 2 x y \frac{d y}{d x} + y = 0$

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Solution

The correct option is C
$- y {(\frac{d y}{d x})}^{2} + 2 x \frac{d y}{d x} = - y$

Equation of family of parabolas with focus at
(0,0) and x - axis as axis is $y^{2} = 4 a (x + a) . . . . . . (i)$
Differentiating (i) with respect to x,
$2 y y_{1} = 4 a; y^{2} = 2 y y_{1} (x + \frac{y y_{1}}{2}) y = 2 x y_{1} + y y_{1}^{2} \Rightarrow y {(\frac{d y}{d x})}^{2} + 2 x \frac{d y}{d x} = y .$

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[EAMCET 2003]