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JEE Main & Advanced Mathematics Differential Equations Question Bank Variable Separable type differential equations

  • question_answer
    The solution of \[{{e}^{dy/dx}}=(x+1)\], \[y(0)=3\] is [Kerala (Engg.) 2005]

    A)                 \[y=x\log x-x+2\]

    B)             \[y=(x+1)\log |x+1|-x+3\]

    C)                 \[y=(x+1)\log |x+1|+x+3\]

    D)                 \[y=x\log x+x+3\]

    E)                 \[y=-(x+1)\log |x+1|+x+3\]

    Correct Answer: B

    Solution :

               \[\frac{dy}{dx}=\log (x+1)\] Þ \[dy=\log (x+1)dx\]                    \[y=\int{\log (x+1)dx}=x.\log (x+1)-\int{\frac{x}{x+1}dx}\]                     \[=x.\log (x+1)-\int{\left( 1-\frac{1}{x+1} \right)}\,dx\]                     \[=x.\log (x+1)-x+\log (x+1)+c=(x+1)\log (x+1)-x+c\]                    \[x=0\] at \[y=3\]                    \[3=(1)\log (1)-0+c\] Þ \[3=0+c\] Þ            \[c=3\]                                 \ \[y=(x+1)\log |x+1|-x+3\].


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