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Continuity And Differentiability

Question
CBSEENMA12035408
Wired Faculty App

If y=ea xsin bx, prove thatfraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight a dy over dx plus left parenthesis straight a squared plus straight b squared right parenthesis straight y equals 0

Solution
Let space space space space space straight y equals straight e to the power of straight a space straight x end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 dy over dx equals straight e to the power of straight a space straight x end exponent left parenthesis straight b space cos space bx right parenthesis plus sin space bx. left parenthesis straight a space straight e to the power of straight a space straight x end exponent right parenthesis therefore space dy over dx equals straight e to the power of straight a space straight x end exponent left parenthesis straight a space sin space bx plus straight b space cos space straight x space bx right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight e to the power of straight a space straight x end exponent left parenthesis straight a space straight b space cos space bx minus straight b squared space sin space bx right parenthesis plus left parenthesis straight a space sin space bx plus straight b space cos space bx right parenthesis space straight a space straight e to the power of straight a space straight x end exponent space space space space space space space space space space space space space space space equals straight e to the power of straight a space straight x end exponent left square bracket straight a space straight b space cos space bx minus straight b squared space sin space bx plus space straight a squared space sin space bx plus straight a space straight b space cos space bx right square bracket therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight e to the power of straight a space straight x end exponent left square bracket 2 space straight a space straight b space cos space bx plus straight a squared space sin space bx minus straight b squared space sin space bx 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... left parenthesis 3 right parenthesis space space space space space space straight L. straight H. straight S equals fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight a dy over dx plus left parenthesis straight a squared plus straight b squared right parenthesis straight y space space space space space space space space space space space space space space space space space equals straight e to the power of straight a space straight x end exponent left square bracket 2 space straight a space straight b space cos space bx plus straight a squared space sin space bx minus straight b squared space sin space bx 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 minus 2 space straight e to the power of straight a space straight x end exponent left parenthesis straight a space sin space bx plus straight b space cos space straight x space bx right parenthesis plus left parenthesis straight a squared plus straight b squared right parenthesis. straight e to the power of straight a space straight x end exponent space sin space bx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 space space space space space space space space space space space space equals straight e to the power of straight a space straight x end exponent left square bracket 2 space straight a space straight b space cos space bx plus straight a squared space sin space bx minus straight b squared space sin space bx minus 2 straight a 2 space sin space bx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus space 2 straight a space straight b space cos space bx plus straight a squared space sin space bx plus straight b squared space sin space bx right square bracket space space space space space space space space space space space space space space space space space equals straight e to the power of straight a space straight x end exponent left parenthesis 0 right parenthesis equals 0 equals straight R. straight H. straight S.

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