If y = sin (log x), prove that $straight x 2 fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight y equals 0.$

space space space space space space space space space space space space space space space straight y equals sin left parenthesis log space straight x right parenthesis therefore space space space space space dy over dx equals cos left parenthesis log space straight x.1 over straight x right parenthesis therefore space space space space space dy over dx equals fraction numerator cos left parenthesis log space straight x right parenthesis over denominator straight x end fraction therefore space space space straight x dy over dx equals cos left parenthesis log space straight x right parenthesis Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma space space space space space space space straight x 2 fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx.1 equals negative sin left parenthesis log space straight x right parenthesis.1 over straight x therefore space space space straight x 2 fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx equals negative sin left parenthesis log space straight x right parenthesis therefore space space space straight x 2 fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx equals negative straight y therefore space space space straight x 2 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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