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Form the differential equation by eliminating the arbitrary constant from the equation $xy = {ae}^{x} + {be}^{- x} + x^{2}$

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$x \frac{d^{2} y}{{dx}^{2}} - 2 \frac{dy}{dx} + xy = 2$

$x \frac{d^{2} y}{{dx}^{2}} + 2 \frac{dy}{dx} + xy = 2$

$x \frac{d^{2} y}{{dx}^{2}} - 2 \frac{dy}{dx} - xy = 2$

$x \frac{d^{2} y}{{dx}^{2}} + 2 \frac{dy}{dx} - xy + x^{2} = 2$

answer is A.

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Detailed Solution

Given $xy - x^{2} = {ae}^{x} + {be}^{- x} \Rightarrow {xy}_{1} + y - 2 x = {ae}^{x} - {be}^{- x} \Rightarrow {xy}_{2} + y_{1} + y_{1} - 2 = {ae}^{x} + {be}^{- x} \Rightarrow x y_{2} + 2 y_{1} - 2 = x y - x^{2}$

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