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Continuity And Differentiability

Question

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If y=cos(log x)+sin(log x), then prove that x²y₂+x y₁+y=0 where y₁ and y₂are first and second order derivatives.

Solution

space space space space space space space space space space space space space space space space space space space space straight y equals cos left parenthesis log space straight x right parenthesis plus 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... left parenthesis 1 right parenthesis therefore space space space space space space space space space space space space space space space straight y subscript 1 equals negative fraction numerator sin left parenthesis log space straight x right parenthesis over denominator straight x end fraction plus fraction numerator cos left parenthesis log space straight x right parenthesis over denominator straight x end fraction rightwards double arrow space space space space space space space space space space space straight x space straight y subscript 1 equals negative sin left parenthesis log space straight x right parenthesis plus cos left parenthesis log space straight x right parenthesis again space differentiating space straight w. straight r. straight t. straight x comma space we space get comma space space space space space space space space space space space space space space space space space space straight x space straight y squared plus 1. straight y to the power of 1 equals negative fraction numerator cos left parenthesis log space straight x right parenthesis over denominator straight x end fraction minus fraction numerator sin left parenthesis log space straight x right parenthesis over denominator straight x end fraction or space space space space space space space space space space space space space space space straight x squared straight y subscript 2 plus xy subscript 1 equals negative cos left parenthesis log space straight x right parenthesis minus sin left parenthesis log space straight x right parenthesis or space space space space space space space space space straight x squared straight y subscript 2 plus xy subscript 1 plus straight y 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 left square bracket because space of space left parenthesis 1 right parenthesis right square bracket therefore space space space straight x squared straight y subscript 2 plus xy subscript 1 plus straight y plus straight y equals 0

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