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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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The equation of the family of curves is
x y = A e^x + B e^–x + x^{2 ...(1)}
Differentiating both sides w.r.t. x, we get,

straight x dy over dx plus straight y space.1 space equals space Ae to the power of straight x minus Be to the power of negative straight x end exponent plus 2 straight x

Again differentiating both sides w.r.t. x, we get,

straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx.1 plus dy over dx space equals space straight A space straight e to the power of straight x plus Be to the power of negative straight x end exponent plus 2

straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus 2 dy over dx space equals space xy space minus straight x squared plus 2

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

which is required differential equation.

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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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