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Continuity And Differentiability

Question
CBSEENMA12035406
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If y = a cos (log x) + b sin (log x), show that straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight y equals 0.

Solution
straight y equals straight a space cos left parenthesis log space straight x right parenthesis plus straight b space sin left parenthesis log space straight x right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis therefore space dy over dx equals negative straight a space sin left parenthesis log space straight x right parenthesis cross times 1 over straight x plus straight b space cos left parenthesis log space straight x right parenthesis cross times 1 over straight x therefore space straight x dy over dx equals negative straight a space sin left parenthesis log space straight x right parenthesis plus straight b space cos left parenthesis log space straight x right parenthesis Again space differentiating space straight w. straight r. straight t. straight x comma space space space space space space space space straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx.1 equals negative straight a space cos left parenthesis log space straight x right parenthesis cross times 1 over straight x minus straight b space sin left parenthesis log space straight x right parenthesis cross times 1 over straight x therefore space space space space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx equals negative straight a space cos left parenthesis log space straight x right parenthesis plus straight b space sin left parenthesis log space straight x right parenthesis or space space space space space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx equals negative left square bracket straight a space cos left parenthesis log space straight x right parenthesis plus straight b space sin left parenthesis log space straight x right parenthesis right square bracket or space space space space space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx equals negative straight y space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket or space space space space space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight y equals 0

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