The solution of x2dydx=4−y2 is

x2dydx=4−y2⇒14−y2dy=1x2dx⇒∫14−y2dy=∫1x2dx⇒Sin−1⁡y2=−1x+c Sin−1⁡(y/2)+1/x=c

The solution of x2dydx=4−y2 is

x2dydx=4−y2⇒14−y2dy=1x2dx⇒∫14−y2dy=∫1x2dx⇒Sin−1⁡y2=−1x+c Sin−1⁡(y/2)+1/x=c

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The solution of $x^{2} \frac{d y}{d x} = \sqrt{4 - y^{2}}$ is

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${Cos}^{- 1} (y / 2) + 1 / x = c$

${Tan}^{- 1} (y / 2) + 1 / x = c$

${Sin}^{- 1} (y / 2) - 1 / x = c$

${Sin}^{- 1} (y / 2) + 1 / x = c$

answer is D.

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Detailed Solution

$x^{2} \frac{d y}{d x} = \sqrt{4 - y^{2}} \Rightarrow \frac{1}{\sqrt{4 - y^{2}}} d y = \frac{1}{x^{2}} d x \Rightarrow \int \frac{1}{\sqrt{4 - y^{2}}} d y = \int \frac{1}{x^{2}} d x \Rightarrow {Sin}^{- 1} (\frac{y}{2}) = - \frac{1}{x} + c {Sin}^{- 1} (y / 2) + 1 / x = c$