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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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The given equation is
y = e^x (a cosx + b sinx) ...(1)
Differentiating both sides w.r.t x, we get

dy over dx equals straight e to the power of straight x left parenthesis straight a space cosx plus space straight b space sinx right parenthesis space plus straight e to the power of straight x left parenthesis negative straight a space sinx space plus space straight b space cosx right parenthesis

dy over dx equals straight y plus straight e to the power of straight x left parenthesis negative straight a space sin space straight x space plus space straight b space cosx right parenthesis

...(2)

open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

Again differentiating w.r.t x,

fraction numerator straight d squared straight y over denominator dx squared end fraction equals space dy over dx plus straight e to the power of straight x left parenthesis negative straight a space sinx space plus space straight b space cosx right parenthesis space plus space straight e to the power of straight x left parenthesis negative straight a space cosx space minus space straight b space sinx right parenthesis

fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space dy over dx plus open parentheses dy over dx minus straight y close parentheses space minus space straight e to the power of straight x left parenthesis straight a space cosx space plus space straight b space sinx right parenthesis 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 2 right parenthesis close square brackets

fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space dy over dx plus dy over dx minus straight y minus straight y

open square brackets because space space of space space left parenthesis 1 right parenthesis close square brackets

fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 dy over dx plus 2 straight y equals 0

, which is required differential equation.

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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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78. Find the differential equation of the family of curves y = a sin (bx + c). a, b. c being arbitrary constants.

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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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