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531.

If space straight x equals log space straight theta comma space straight theta greater than 0 space and space straight y equals 1 over straight theta space then space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx equals 0.

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532.

If space space straight y equals straight e to the power of straight x left parenthesis sin space straight x plus cos space straight x right parenthesis comma space prove space that space dy over dx minus 2 dy over dx plus 2 straight y equals 0

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533.

If space straight y equals 5 space cos space straight x minus 3 space sin space straight x comma space prove space that space fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y equals 0.

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534.

Find space fraction numerator straight d squared straight y over denominator dx squared end fraction comma space if space straight y equals straight e to the power of straight x space sin end exponent space to the power of straight x.

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535.

If space straight y space equals space cos to the power of negative 1 end exponent space straight x comma space find space fraction numerator straight d squared straight y over denominator dx squared end fraction space in space terms space of space straight y space alone.

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536. If y = sin (sin x), prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction plus tan space straight x space dy over dx plus space straight y space cos squared space straight x equals 0

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537. If y = sin (log x), prove that

straight x 2 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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538. f y = a sin 2x + b cos 2x, show that

fraction numerator d squared y over denominator d x squared end fraction plus 4 space y equals 0.

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straight y equals space straight A space sin space straight x plus straight B space cos space straight x space space space... left parenthesis 1 right parenthesis therefore space space space space space space space space dy over dx equals straight A space cos space straight x minus space straight B space sin space straight x therefore space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative straight A space sin space straight x minus space straight B space cos space straight x therefore space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative straight y space space space space space space space space space space space space space space space space space space space space space space space space 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 right square bracket therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y equals 0

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540. If y = A cos n x + B sin n x, prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight n squared space straight y equals 0

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