A) \[y={{\log }_{e}}(x)+c\]
B) \[y={{({{\log }_{e}}x)}^{2}}+c\]
C) \[y=\pm \sqrt{{{({{\log }_{e}}x)}^{2}}+2c}\]
D) \[xy={{x}^{y}}=k\]
Correct Answer: C
Solution :
The given differential equation can be written as \[x={{e}^{xy\frac{dy}{dx}}}\] \[\Rightarrow \] \[\log x=xy\frac{dy}{dx}\Rightarrow ydy=\frac{\log x}{x}dx\] \[\Rightarrow \] \[ydy=\log x\,\,d(\log x)\] On integrating, we get \[\frac{{{y}^{2}}}{2}=\frac{{{(\log x)}^{2}}}{2}+c\] \[\Rightarrow \] \[{{y}^{2}}={{({{\log }_{e}}x)}^{2}}+2c\] \[\Rightarrow \] \[y=\pm \sqrt{{{({{\log }_{e}}x)}^{2}}+2c}\]You need to login to perform this action.
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