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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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The given equation is

open parentheses straight x minus straight a close parentheses squared plus 2 straight y squared space equals space straight a squared

...(1)
Differentiating both sides w.r.t.x , we get.

2 left parenthesis straight x minus straight a right parenthesis plus 4 straight y dy over dx space equals 0

therefore space space

straight x minus straight a equals negative 2 straight y dy over dx

...(2)
and

straight a equals straight x plus 2 straight y dy over dx

...(3)
Putting the values of x - a, a from (2), (3) in (1), we get,

open parentheses negative 2 straight y space dy over dx close parentheses squared plus 2 straight y squared space equals space open parentheses straight x plus 2 straight y dy over dx close parentheses squared

4 straight y squared open parentheses dy over dx close parentheses squared plus 2 straight y squared space equals space straight x squared plus 4 straight y squared open parentheses dy over dx close parentheses squared plus 4 xy dy over dx

2 straight y squared space equals space straight x squared plus 4 xy dy over dx

2 straight y squared minus straight x squared space equals space 4 xy dy over dx

which is the required differential equation of the given family of curves.

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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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