Question

$If space space space space straight y equals left parenthesis straight a plus straight b space straight t right parenthesis space straight e to the power of straight n space straight t end exponent comma space prove space that space fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight n dy over dx plus straight n squared straight y equals 0.$

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Solution

Let space space space space space space space space space space straight y equals left parenthesis straight a plus straight b space straight t right parenthesis space straight e to the power of straight n space straight t end exponent therefore space space space space space space dy over dt equals left parenthesis straight a plus space straight b space straight t right parenthesis left parenthesis straight e to the power of straight n space straight t end exponent right parenthesis plus straight e to the power of straight n space straight t end exponent. straight b space space space space space space space space space space space space space space space space space space space space space space 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 dy over dt equals straight e to the power of straight n space straight t end exponent left square bracket straight a space straight n plus straight b space straight n space straight t plus straight b right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 2 right parenthesis and space fraction numerator straight d squared straight y over denominator dt squared end fraction equals straight e to the power of straight n space straight t end exponent left square bracket straight b space straight n right square bracket plus left parenthesis straight a space straight n plus straight b space straight n space straight t plus straight b right parenthesis. left parenthesis straight n space straight e to the power of straight n space straight t end exponent right parenthesis therefore space space space space fraction numerator straight d squared straight y over denominator dt squared end fraction equals straight e to the power of straight n space straight t end exponent left square bracket straight b space straight n plus straight a space straight n squared plus straight b space straight n squared space straight t plus straight b space straight n right square bracket space space space space space space space space space space space 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 3 right parenthesis space space space straight L. straight H. straight S equals fraction numerator straight d squared straight y over denominator dt squared end fraction minus 2 space straight n space dy over dt plus straight n squared space straight y space space space space space equals straight e to the power of straight n space straight t end exponent left square bracket straight b space straight n plus straight a space straight n squared plus straight b space straight n squared plus straight b space straight n right square bracket minus 2 space straight n. straight e to the power of straight n space straight t end exponent left parenthesis straight a space straight b plus straight b space straight n space straight t plus straight b right parenthesis plus straight n squared. left parenthesis straight a space straight b space straight t right parenthesis straight e to the power of straight n space straight t end exponent space space space space space equals straight e to the power of straight n space straight t end exponent left square bracket straight b space straight n plus straight a space straight n squared plus straight b space straight n squared plus straight b space straight n minus 2 space straight a space straight n squared minus 2 space straight b space straight n squared space straight t minus 2 space straight b space straight n plus space straight a space straight n squared plus straight b space straight n squared space straight t right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis right square bracket space space space space space equals straight e to the power of straight n space straight t end exponent left parenthesis 0 right parenthesis equals 0 equals straight R. straight H. straight S.

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