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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.

The given equation is
y = 2 (x² - 1) + c e^-x² or ye^x² = 2 (x²1) e^x² + c
Differentiating both sides w.r.t. x, we get.
$straight y. straight e to the power of straight x squared end exponent. space 2 space straight x space plus space straight e to the power of straight x squared end exponent. space dy over dx space equals space 2 left parenthesis straight x squared minus 1 right parenthesis space straight e to the power of straight x squared end exponent. space 2 space straight x space plus space 2 straight e to the power of straight x squared end exponent. space space 2 straight x space plus space 0$
or $2 xy plus dy over dx space equals space 4 straight x left parenthesis straight x squared minus 1 right parenthesis space plus space 4 straight x space space space space space or space space space dy over dx space equals space 4 straight x left parenthesis straight x squared minus 1 right parenthesis space plus space 4 straight x space minus space 2 xy$
which is required differential equation.