A) \[y={{\tan }^{-1}}x+c\]
B) \[x={{\tan }^{-1}}y+c\]
C) \[\tan (xy)=c\]
D) \[y-x=c(1+xy)\]
Correct Answer: D
Solution :
[d]: The given differential equation is \[\frac{dy}{dx}=\frac{1+{{y}^{2}}}{1+{{x}^{2}}}\Rightarrow \frac{dy}{1+{{y}^{2}}}=\frac{dx}{1+{{x}^{2}}}\] \[\Rightarrow \int_{{}}^{{}}{\frac{dy}{1+{{y}^{2}}}}=\int_{{}}^{{}}{\frac{dx}{1+{{x}^{2}}}}\] \[\Rightarrow {{\tan }^{-1}}y={{\tan }^{-1}}x+{{\tan }^{-1}}c\] \[\Rightarrow {{\tan }^{-1}}\left( \frac{y-x}{1+xy} \right)={{\tan }^{-1}}c\] \[\Rightarrow \frac{y-x}{1+xy}=c\Rightarrow y-x=c(1+xy)\], which is the required solution of the given differential equation.
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