Find the differential equation of the family of curves  y = A s
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    Differential Equations

    Multiple Choice QuestionsShort 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.

    The given equation is
                y = A sin mx + B cos mx.                               ...(1)
    therefore space space space dy over dx equals space straight m space straight A space sin space mx space plus space straight m space straight B space cos space mx

    rightwards double arrow space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space straight m squared straight A space cos space mx space minus space straight m squared space straight B space sin space mx rightwards double arrow space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space minus straight m squared left parenthesis straight A space cos space mx space plus space straight B space sin space mx right parenthesis rightwards double arrow space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals space minus straight m squared straight y space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets rightwards double arrow space space space fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight m squared straight y space equals 0

                             

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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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    Multiple Choice QuestionsLong 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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    Multiple Choice QuestionsShort 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.
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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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