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61. Form the differential equation representing the family of curves y = a cos (x + b) where a and b are arbitrary constants.

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62. Find the differential equation of the family of curves y = A sin mx + B cos mx. where m is fixed, and A and B are arbitrary constants.

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63. Form the differential equation corresponding to y² = m (a² – x²) by eliminating m and a.

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The given equation is

straight y squared space equals straight a space left parenthesis straight b minus straight x right parenthesis space left parenthesis straight b plus straight x right parenthesis space space or space space straight y squared space equals straight a space left parenthesis straight b squared minus straight x squared right parenthesis

...(1)

therefore space space space space space 2 straight y dy over dx space equals space straight a left parenthesis negative 2 straight x right parenthesis space or space straight y dy over dx space equals space minus ax

...(2)

Again

straight y fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx. dy over dx space equals space minus straight a

...(3)
Eliminating a from (2) and (3), we get,

straight y dy over dx space equals space xy fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x open parentheses dy over dx close parentheses squared space space or space space straight x space straight y fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x open parentheses dy over dx close parentheses squared minus straight y dy over dx equals 0

which is the required differential equation.

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65. Form the differential equation of the family of curves represented by the equation (x – a)² + 2 y² = a², a being the parameter.

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66. Prove that x² – y² = c (x² + y² )² is the general solution of differential equation (x³ – 3 x y² ) dx = (y³ –3 x² y) dy . where c is a parameter.

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67. Form a differential equation from the equation y = 2(x² - 1) + ce^-x².

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68. Form the differential equation of the family of curves

straight y equals Ax plus straight B over straight x

where A and B are constants.

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69. Form the differential equation of the family of curves

straight y equals Ae to the power of Bx

where A and B are constants.

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70. Form the differential equation of the family of curves

straight y equals Ae to the power of straight x plus Be to the power of negative straight x end exponent

where A and B are constants.

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