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Continuity And Differentiability

Question

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

Solution

Here space space space space straight y equals straight A space cos space nx space plus space straight B space sin space straight n space straight x 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 space space dy over dx equals negative space straight A space straight n space sin space straight n space straight x plus straight B space straight n space cos space straight n space straight x therefore space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative straight A space straight n squared space cos space straight n space straight x minus space straight B space straight n squared space sin space straight n space straight x rightwards double arrow space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative straight n squared left square bracket straight A space cos space straight n space straight x plus straight B space sin space straight n space straight x right square bracket rightwards double arrow space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative straight n squared space 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 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 rightwards double arrow space space fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight n squared space straight y equals 0.

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