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J & K CET Engineering J and K - CET Engineering Solved Paper-2012

  • question_answer
    The trajectory of the differential equation \[\frac{dx}{dt}=rx\left( 1-\frac{x}{k} \right),\,\,\,\,r>0\] is  monotonically increasing, if

    A)  \[0<x<k\]

    B)  \[x(0)>k\]

    C)  \[\frac{k}{2}<x(0)<2k\]

    D)  None of these

    Correct Answer: A

    Solution :

    \[\therefore \] Curve trajectory \[=\frac{dx}{dt}\] \[=-rx\left( \frac{x}{k}-1 \right)\] For monotonically increasing, \[\frac{dx}{dt}>0\] \[\Rightarrow \] \[-rx\left( \frac{x}{k}-1 \right)>0\] \[\Rightarrow \] \[x(x-k)<0\] \[\Rightarrow \] \[0<x<k\]


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