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Continuity and Differentiability

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551. If y=e^{a x}sin bx, prove that $fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight a dy over dx plus left parenthesis straight a squared plus straight b squared right parenthesis straight y equals 0$

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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552.

If space straight y equals straight x space log open parentheses fraction numerator straight x over denominator straight a plus straight b space straight x end fraction close parentheses comma space prove space that space straight x cubed fraction numerator straight d squared straight y over denominator dx squared end fraction equals open parentheses straight x dy over dx minus straight y squared close parentheses.

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553.

If space straight x equals tan open parentheses 1 over straight a log space straight y close parentheses comma space show space that space left parenthesis 1 plus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus left parenthesis 2 space straight x minus straight a right parenthesis dy over dx equals 0.

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554. If y=x^x, prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction minus 1 over straight y open parentheses dy over dx close parentheses squared minus straight y over straight x equals 0.

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555.

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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556.

If space straight y equals fraction numerator straight a space straight x plus straight b over denominator straight c space straight x plus straight d end fraction comma space prove space that space 2 straight y subscript 1 space straight y subscript 3 equals 3 space straight y subscript 2 squared.

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557.

If space straight y equals log space left parenthesis 1 plus cos space straight x right parenthesis comma space prove space that space straight y subscript 1 space straight y subscript 2 plus straight y subscript 3 equals 0.

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558.

If space 2 space straight y equals straight x open parentheses 1 plus dy over dx close parentheses comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction space is space constant space.

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559. If y = (tan^-1 x)², then prove that (1 + x²)² y₂ + 2x(1 + x²)y_{1 = 2.}

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560.

If space straight y equals sin to the power of negative 1 end exponent straight x comma space then space show space that space open parentheses 1 minus straight x squared close parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x dy over dx equals 0.

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551. If y=ea xsin bx, prove that

551. If y=e^{a x}sin bx, prove that $fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight a dy over dx plus left parenthesis straight a squared plus straight b squared right parenthesis straight y equals 0$