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The differential equation corresponding to the family of parabola $y^{2} = 4 a (x + a)$ where a is the parameter, is

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

$y {(\frac{dy}{dx})}^{2} + 2 x (\frac{dy}{dx}) - y = 0$

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

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

answer is C.

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Detailed Solution

$y^{2} = 4 a (x + a) y^{2} = 4 ax + 4 a^{2} \to (1) 2 y \frac{dy}{dx} = 4 a y \frac{dy}{dx} = 2 a (1) \Rightarrow y^{2} = 2 y \frac{dy}{dx} x + y^{2} {(\frac{dy}{dx})}^{2} \Rightarrow y {(\frac{dy}{dx})}^{2} + 2 x \frac{dy}{dx} - y = 0$

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