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Continuity And Differentiability

Question

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If y=A sin x+B cos x, prove that $fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y equals 0$

Solution

straight y equals space straight A space sin space straight x plus straight B space cos space straight x space space space... left parenthesis 1 right parenthesis therefore space space space space space space space space dy over dx equals straight A space cos space straight x minus space straight B space sin space straight x therefore space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative straight A space sin space straight x minus space straight B space cos space straight x therefore space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction 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 left square bracket because space of space left parenthesis 1 right parenthesis right square bracket therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y equals 0

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