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71. Form the differential equation of the family of curves by eliminating arbitrary constants a and b.
y = a e^3x + be^-2x

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72. Form the differential equation of the family of curves by eliminating arbitrary constants a and b.
y = e^2x (a + b x)

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73. Form the differential equation of the family of curves by eliminating arbitrary constants a and b.
y = e^x (a cosx + b sinx)

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74. Form a differential equation from the equation y = ae ^2x + be^{– 3x} , a , b being constants.

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75. Form the differential equation of the following family of curves:
x y = A e^x + B e^–x + x²

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76. Find the differential equation of the family of curves y = ae^x + be^2x + ce^3x, where a, b. c are arbitrary constants.

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77.

Obtain the differential equation from the equation y = e^x (a cos 2x + b sin 2x). where a and b are arbitrary constants.

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The given equation is
y = a sin (bx + c) ...(1)
Differentiating w.r.t.x, successively three times,

dy over dx space equals straight b space straight a space cos left parenthesis bx plus straight c right parenthesis

...(2)

fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space minus straight b squared straight a space sin left parenthesis bx plus straight c right parenthesis

...(3)

fraction numerator straight d cubed straight y over denominator dx cubed end fraction space equals space minus straight b cubed straight a space cos left parenthesis bx plus straight c right parenthesis

...(4)
From (1) and (2), (3) and (4), we get,

straight y fraction numerator straight d cubed straight y over denominator dx cubed end fraction equals negative straight b cubed straight a squared space sin space left parenthesis bx plus straight c right parenthesis space cos space left parenthesis bx plus straight c right parenthesis

equals space left square bracket straight b space straight a space cos space left parenthesis bx plus straight c right parenthesis right square bracket space left square bracket straight b squared straight a space sin left parenthesis bx plus straight c right parenthesis right square bracket

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

is the required differential equation.

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79. Form the differential equation of the family of circles having centre on x-axis and passing through the origin.

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80. Determine the differential equation that will represent the family of all circles having centres on the x-axis and radius unity.

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