JEE Main & Advanced Mathematics Differential Equations Question Bank Self Evaluation Test - Differential Equations

  • question_answer
    What is the solution of the equation\[\ln \,\left( \frac{dy}{dx} \right)+x=0\]?

    A) \[y+{{e}^{x}}=c\]

    B) \[y-{{e}^{-x}}=c\]

    C) \[y+{{e}^{-x}}=c\]

    D) \[y-{{e}^{x}}=c\]

    Correct Answer: C

    Solution :

    [c] Consider the given differential equation \[\ln \,\left( \frac{dy}{dx} \right)+x=0\Rightarrow ln\left( \frac{dy}{dx} \right)=-x\] \[\Rightarrow \frac{dy}{dx}={{e}^{-x}}\] On separating the variables, we get \[dy={{e}^{-x}}dx,\] On integrating both sides, we get \[\int{dy=\int{{{e}^{-x}}dx}}\] \[\Rightarrow y=\frac{{{e}^{-x}}}{-1}+C=-{{e}^{-x}}+C\Rightarrow y+{{e}^{-x}}=C\]

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