The D.E whose solution is $y = c (x - c)^{2}$ is

{(\frac{d y}{d x})}^{3} = 4 y (x \frac{d y}{d x} - 2 y)

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y {_{1}}^{3} = 2 y (x y_{1} - y)

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{(\frac{d y}{d x})}^{3} = 4 y (2 x \frac{d y}{d x} - y)

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{(\frac{d y}{d x})}^{3} = 2 y (x \frac{d y}{d x} - 4 y)

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Solution

The correct option is A ${(\frac{d y}{d x})}^{3} = 4 y (x \frac{d y}{d x} - 2 y)$
$y = c (x - c)^{2}$
$\Rightarrow y_{1} = 2 c (x - c)$
$\Rightarrow c (x - c) = y_{1} / 2$
$\Rightarrow y = (x - c) y_{1} / 2$
$∴ x - \frac{2 y}{y_{1}} = c$
$s u b s t i t u t e t h i s i n f i r s t s t e p$

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