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space space space space space space space space space space space straight x equals log space straight theta comma space space space straight y equals 1 over straight theta therefore space dx over dθ equals 1 over straight theta comma space dy over dθ equals negative 1 over straight theta squared space space space space space space dy over dx equals fraction numerator begin display style dy over dθ end style over denominator begin display style dx over dθ end style end fraction equals negative 1 over straight theta squared cross times straight theta over 1 equals negative 1 over straight theta equals negative straight theta to the power of negative 1 end exponent therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative left parenthesis negative 1 right parenthesis straight theta to the power of negative 2 end exponent dθ over dx equals 1 over straight theta squared. straight theta equals 1 over straight theta space space space space straight L. straight H. straight S equals fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx equals 1 over straight theta minus 1 over straight theta equals 0 space space space space space space space space space space space space space space space equals straight R. straight H. straight S.

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

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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