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JEE Main & Advanced Mathematics Differentiation Question Bank Differentiation by Substitution

  • question_answer
    If \[y=a\cos \,(\log x)+b\sin \,(\log x)\] where \[a,\,b\] are parameters then \[{{x}^{2}}{y}''\,+\,x{y}'\,=\] [EAMCET 2002]

    A)            \[y\]

    B)            \[-y\]

    C)            \[2y\]

    D)            \[-2y\]

    Correct Answer: B

    Solution :

               \[y=a\cos (\log x)+b\sin (\log x)\]            Þ  \[y'=\frac{-\,a\sin (\log x)}{x}+\frac{b\cos (\log x)}{x}\]            Þ  \[xy'=-\,a\sin (\log x)+b\cos (\log x)\]            Þ  \[x{y}''+{y}'=\frac{-a\cos (\log x)}{x}-\frac{b\sin (\log x)}{x}\]            Þ  \[{{x}^{2}}{y}''+x{y}'=-[a\cos (\log x)+b\sin (\log x)]\]            Þ \[{{x}^{2}}y''\,+\,xy'=-y.\]


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