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Manipal Engineering Manipal Engineering Solved Paper-2010

  • question_answer
    Solution of the differential equation                 \[x=1+xy\frac{dy}{dx}+\frac{{{(xy)}^{2}}}{2!}{{\left( \frac{dy}{dx} \right)}^{2}}\]\[+\frac{{{(xy)}^{3}}}{3!}{{\left( \frac{dy}{dx} \right)}^{3}}+...\]is

    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                 \[\frac{{{y}^{2}}}{2}=\frac{{{(\log x)}^{2}}}{2}+c\] \[\Rightarrow \]               \[{{y}^{2}}={{({{\log }_{e}}x)}^{2}}+2c\] \[\Rightarrow \]               \[y=\pm \sqrt{{{({{\log }_{e}}x)}^{2}}+2c}\]


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