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Continuity And Differentiability

Question

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f y = a sin 2x + b cos 2x, show that $fraction numerator d squared y over denominator d x squared end fraction plus 4 space y equals 0.$

Solution

Here space space straight y equals straight a space sin space 2 straight x space plus straight b space cos space 2 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... left parenthesis 1 right parenthesis therefore space space space space dy over dx equals 2 space straight a space cos space 2 straight x minus 2 space straight b space sin space 2 straight x therefore space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative 4 space straight a space sin space 2 straight x minus 4 space straight b space cos space 2 straight x rightwards double arrow space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative 4 left parenthesis straight a space sin space 2 straight x plus straight b space cos space 2 straight x right parenthesis rightwards double arrow space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative 4 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 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 4 space straight y equals 0.

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