Question

$If space straight y equals 5 space cos space straight x minus 3 space sin space straight x comma space prove space that space fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y equals 0.$

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Solution

space space space space space space space space space space space space space space space space space space straight y equals 5 space cos space straight x minus 3 space sin space straight x space space space space space space space space space space space space space dy over dx equals negative 5 space sin space straight x minus space 3 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 5 space cos space straight x plus space 3 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 open parentheses 5 space cos space straight x minus space 3 space sin space straight x close parentheses 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 therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y equals 0

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