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JEE Main & Advanced Mathematics Differential Equations Question Bank Exact defferential equations

  • question_answer
    If \[xdy=y\,(dx+ydy),\,y>0\] and \[y(1)=1,\] then \[y(-3)\] is equal to             [IIT Screening 2005]

    A)                 1             

    B)                 3

    C)                 5             

    D)                 ?1

    Correct Answer: B

    Solution :

                       \[xdy=y(dx+ydy)\] Þ \[\frac{xdy-ydx}{{{y}^{2}}}=dy\] Þ \[-d\left( \frac{x}{y} \right)=dy\]                    Integrating both sides, we get \[\frac{x}{y}+y=c\]                    \[\because \]\[y(1)=1\Rightarrow \,c=2\];  \[\therefore \]\[\frac{x}{y}+y=2\]                    For \[x=-3\],                                 \[{{y}^{2}}-2y-3=0\Rightarrow y=-1\] or 3 \[\Rightarrow y=3\]                \[(\because \,\,y>0)\]


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