The solution of 1+x2dx+1+y2dy=0 is

Given equation is 1+x2dx+1+y2dy=0⇒∫1+x2dx+∫1+y2dy=0⇒x21+x2+12Sinh−1⁡x+y21+y2+12Sinh−1⁡y=c2⇒x1+x2+y1+y2+Sinh−1⁡x+Sinh−1⁡y=c

The solution of 1+x2dx+1+y2dy=0 is

Given equation is 1+x2dx+1+y2dy=0⇒∫1+x2dx+∫1+y2dy=0⇒x21+x2+12Sinh−1⁡x+y21+y2+12Sinh−1⁡y=c2⇒x1+x2+y1+y2+Sinh−1⁡x+Sinh−1⁡y=c

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The solution of $\sqrt{1 + x^{2}} dx + \sqrt{1 + y^{2}} dy = 0$ is

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$x \sqrt{1 + x^{2}} + y \sqrt{1 + y^{2}} + {Sinh}^{- 1} x + {Sinh}^{- 1} y = c$

$x \sqrt{1 + x^{2}} + y \sqrt{1 + y^{2}} = c$

$x \sqrt{1 + x^{2}} - y \sqrt{1 + y^{2}} + {Sinh}^{- 1} x - {Sinh}^{- 1} y = c$

${Sinh}^{- 1} x + {Sinh}^{1} y = c$

answer is A.

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Detailed Solution

Given equation is $\sqrt{1 + x^{2}} dx + \sqrt{1 + y^{2}} dy = 0 \Rightarrow \int \sqrt{1 + x^{2}} dx + \int \sqrt{1 + y^{2}} dy = 0$

$\begin{array}{l} \Rightarrow \frac{x}{2} \sqrt{1 + x^{2}} + \frac{1}{2} {Sinh}^{- 1} x + \frac{y}{2} \sqrt{1 + y^{2}} + \frac{1}{2} {Sinh}^{- 1} y = \frac{c}{2} \\ \Rightarrow x \sqrt{1 + x^{2}} + y \sqrt{1 + y^{2}} + {Sinh}^{- 1} x + {Sinh}^{- 1} y = c \end{array}$