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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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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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Let centre on x-axis be (a. 0). Also radius = 1
∴ the equation of the family of circles is

open parentheses straight x minus straight a close parentheses squared plus left parenthesis straight y minus 0 right parenthesis squared space equals space 1 space space space space space space space space space space space space or space space space space space space left parenthesis straight x minus straight a right parenthesis squared plus straight y squared space equals 1 space

...(1)
Differentiating w.r.t x, we get,

2 left parenthesis straight x minus straight a right parenthesis plus 2 straight y dy over dx space equals 0

straight x minus straight a space equals space minus straight y dy over dx

Putting this value of x - a in (1), we get,

open parentheses negative straight y dy over dx close parentheses squared plus straight y squared space equals space 1 space space space space space space space space space space space space space space space space space space space space or space space space space space space space straight y squared open parentheses dy over dx close parentheses squared plus straight y squared space equals space 1

which is required differential equation.

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