A) \[\frac{x}{1+{{x}^{2}}}\]
B) \[\frac{1}{2}\log (1+{{x}^{2}})\]
C) \[\sqrt{1+{{x}^{2}}}\]
D) \[x\]
E) \[\frac{1}{1+{{x}^{2}}}\]
Correct Answer: C
Solution :
\[(1+{{x}^{2}})\frac{dy}{dx}+xy=x\] \[\Rightarrow \] \[\frac{dy}{dx}+\frac{x}{(1+{{x}^{2}})}y=\frac{x}{(1-{{x}^{2}})}\] \[IF={{e}^{\int{\frac{x}{1+{{x}^{2}}}dx}}}\] \[={{e}^{\frac{1}{2}\log (1+{{x}^{2}})}}\] \[={{e}^{\log \sqrt{1+{{x}^{2}}}}}\] \[=\sqrt{1+{{x}^{2}}}\]
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