A) \[{{x}^{\log x}}\]
B) \[{{(\sqrt{x})}^{\log x}}\]
C) \[{{(\sqrt{e})}^{{{(\log x)}^{2}}}}\]
D) \[{{e}^{{{x}^{2}}}}\]
E) \[\frac{{{x}^{2}}}{2}\]
Correct Answer: C
Solution :
Given differential equation is \[x\frac{dy}{dx}+y\log x=x{{e}^{x}}{{x}^{-1/2\log x}}\] \[\frac{dy}{dx}+y\frac{1}{x}\log x={{e}^{x}}{{x}^{-1/2\log x}}\] Here, \[P=\frac{1}{x}\log x\]and\[Q={{e}^{x}}{{x}^{-1/2\log x}}\] \[\therefore \] \[IF={{e}^{\int{\frac{1}{x}\log x\,dx}}}\] \[={{e}^{\frac{{{(\log x)}^{2}}}{2}}}\] \[={{(\sqrt{e})}^{{{(\log x)}^{2}}}}\]
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