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Continuity and Differentiability

Short Answer Type

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

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

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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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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