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Explain solution for  RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple choice Question Question 31 maths textbook solution.

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Answer : \frac{x+y}{1-x y}=C

Hint : Use variable separable equation i.e. take the y terms in one side and x terms to the other.

Given : \left(x^{2}+1\right) \frac{d y}{d x}+\left(y^{2}+1\right)=0

Explanation : \left(x^{2}+1\right) \frac{d y}{d x}=-\left(y^{2}+1\right)

\Rightarrow \frac{dy}{y^{2}+1}=-\frac{dx}{x^{2}+1}

integrate both sides

\begin{aligned} &\Rightarrow \tan ^{-1} y=-\tan ^{-1} x+\tan ^{-1} C \\ &\Rightarrow \tan ^{-1} y+\tan ^{-1} x=\tan ^{-1} C \\ &\Rightarrow \tan ^{-1}\left[\frac{x+y}{1-x y}\right]=\tan ^{-1} C \\ &\Rightarrow \frac{x+y}{1-x y}=C \end{aligned}

Note: Final answer is not matching

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