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541. If y = 3e^2x + 2e^3x, then prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction minus 5 dy over dx plus 6 straight y equals 0.

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542.

If space straight y space equals straight A space straight e to the power of straight m space straight x end exponent plus straight B space straight e to the power of straight n space straight x end exponent comma space show space that space dy over dx minus left parenthesis straight m plus straight n right parenthesis dy over dx plus straight m space straight n space straight y equals 0

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543.

If space straight y equals straight a space straight e to the power of mx plus straight b space straight e to the power of negative mx end exponent comma space prove space that space fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight m squared space straight y equals 0

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544.

If space straight y equals 500 space straight e to the power of 7 straight x end exponent plus 600 space straight e to the power of negative 7 straight x end exponent comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction equals 49 space straight y.

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545.

If space straight e to the power of straight y left parenthesis straight x plus 1 right parenthesis equals 1 comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction equals open parentheses dy over dx close parentheses squared

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546. If y=sin ^-1x, prove that

dy over dx equals straight x over open parentheses 1 minus straight x squared close parentheses to the power of begin display style 3 over 2 end style end exponent

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space space space space space space space space space space space space straight y equals left parenthesis log space straight x right parenthesis squared therefore space space space dy over dx equals 2. log space straight x cross times 1 over straight x equals 2 fraction numerator log space straight x over denominator straight x end fraction therefore space straight x dy over dx equals 2 space log space straight x Diffe. space again space straight w. straight r. straight t. straight x comma straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx.1 equals 2 over straight x rightwards double arrow space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx equals 2 space space rightwards double arrow space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx minus 2 equals 0

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548. If y=cos(log x)+sin(log x), then prove that x²y₂+x y₁+y=0 where y₁ and y₂are first and second order derivatives.

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549. If =3 cos(log x)+4 sin(log x), show that x²y₂+x y₁+y=0.

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550. If y = a cos (log x) + b sin (log x), show that

straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight y equals 0.

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