Form the differential equation by eliminating the arbitrary constant from the equation y = a(x - a)².

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${(\frac{d y}{d x})}^{3} + 4 x y \frac{d y}{d x} - 8 y^{2} = 0$

${(\frac{d y}{d x})}^{3} - 4 x y \frac{d y}{d x} - 8 y^{2} = 0$

${(\frac{dy}{dx})}^{3} + 4 xy \frac{dy}{dx} + 8 y^{2} = 0$

${(\frac{dy}{dx})}^{3} - 4 xy \frac{dy}{dx} + 8 y^{2} = 0$

answer is A.

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Detailed Solution

Given that $y = a (x - a)^{2} \to (1) \Rightarrow \frac{d y}{d x} = 2 a (x - a) \to (2)$

$\Rightarrow {(\frac{d y}{d x})}^{2} = 4 a^{2} (x - a)^{2} = 4 a y \Rightarrow a = \frac{1}{4 y} {(\frac{d y}{d x})}^{2}$

$(2) \Rightarrow \frac{d y}{d x} = 2 [\frac{1}{4 y} {(\frac{d y}{d x})}^{2}] [x - \frac{1}{4 y} {(\frac{d y}{d x})}^{2}] \Rightarrow 8 y^{2} = \frac{d y}{d x} [4 x y - {(\frac{d y}{d x})}^{2}] \Rightarrow {(\frac{d y}{d x})}^{3} - 4 x y \frac{d y}{d x} + 8 y^{2} = 0$