A) \[{{x}^{\log x}}\]
B) \[{{(\sqrt{x})}^{\log x}}\]
C) \[{{(\sqrt{e})}^{\log x}}\]
D) \[{{e}^{{{x}^{2}}}}\]
E) \[{{x}^{2}}/2\]
Correct Answer: B
Solution :
\[\frac{dy}{dx}+\left( \frac{\log x}{x} \right)y={{e}^{x}}{{x}^{-\frac{1}{2}\log x}}\] I.F.\[={{e}^{\int{\frac{\log x}{x}dx}}}={{e}^{\frac{1}{2}{{(\log x)}^{2}}}}={{\left( {{e}^{\frac{1}{2}(\log x)}} \right)}^{\log x}}\] \[={{\left( {{e}^{\log \sqrt{x}}} \right)}^{\log x}}={{(\sqrt{x})}^{\log x}}\]
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