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Differential Equations

Short Answer Type

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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Long Answer Type

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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Short Answer Type

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.

The given equation is
$straight y equals Ax plus straight B over straight x$ ...(1)
$therefore space space space dy over dx space equals space straight A minus straight B over straight x squared$ ...(2)
$rightwards double arrow space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space fraction numerator 2 straight B over denominator straight x cubed end fraction$ ...(3)
From (3), B = $straight x cubed over 2 fraction numerator straight d squared straight y over denominator dx squared end fraction$
From (2), $dy over dx space equals space straight A minus straight x over 2 fraction numerator straight d squared straight y over denominator dx squared end fraction space space rightwards double arrow space space space straight A space equals space straight x over 2 fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx$
Putting value of A and B in (1), we get,
$straight y space equals straight x squared over 2 fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight x squared over 2 fraction numerator straight d squared straight y over denominator dx squared end fraction space space space or space space space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx minus straight y space equals 0$
which is the required differential equation.