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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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64. Form the differential equation corresponding to y² = a (b – x) (b + x) by eliminating a and b.

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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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The given equation is

straight y space equals space straight A space straight e to the power of Bx

...(1)

therefore space space space space dy over dx space equals space straight A space straight B space straight e to the power of Bx space space space space space space space space or space space space space space dy over dx space equals space straight B space straight y

...(2)

open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

therefore space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space straight B dy over dx

...(3)
From (2) and (3),

dy over dx space equals space fraction numerator straight y begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style over denominator begin display style dy over dx end style end fraction

straight y fraction numerator straight d squared straight y over denominator dx squared end fraction space equals open parentheses dy over dx close parentheses squared

which is the required differential equation.

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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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