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If C is parameter then the differential equation whose solution is $y = C^{2} + \frac{C}{x}$ is

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

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

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

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

answer is C.

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Detailed Solution

$y = c^{2} + \frac{c}{x} \Rightarrow y_{1} = \frac{- c}{x^{2}} \Rightarrow c = - x^{2} y_{1} ∴ y = {(- x^{2} y_{1})}^{2} + \frac{(- x^{2} y_{1})}{x} \Rightarrow y = x^{4} y_{1}^{2} - x y_{1}$

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