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The equation of family of curves is
y = a e^3x + be^-2x ...(1)Differentiating both sides w.r.t.x, we get

straight y subscript 1 space equals 3 space straight a space straight e to the power of 3 straight x end exponent space minus space 2 space straight b space straight e to the power of negative 2 straight x end exponent

...(2)
Again differentiating w.r.t. x, we get,

straight y subscript 2 space equals space 9 space straight a space straight e to the power of 3 straight x end exponent plus 4 space be to the power of negative 2 straight x end exponent

...(3)
Mutliplying (2) by 3 subtracting from (3), we get,

straight y subscript 2 minus 3 straight y subscript 1 space equals space 10 space straight b space straight e to the power of negative 2 straight x end exponent

therefore space space space space straight b space straight e to the power of negative 2 straight x end exponent space equals space 1 over 10 left parenthesis straight y subscript 2 minus 3 straight y subscript 1 right parenthesis

...(4)
Multiplying (2) by 2 and adding it to (3), we get,

straight y subscript 2 plus 2 straight y subscript 1 space equals space 15 space straight a space straight e to the power of 3 straight x end exponent

therefore space space space space space space space space ae to the power of 3 straight x end exponent space equals space 1 over 15 left parenthesis straight y subscript 2 plus 2 straight y subscript 1 right parenthesis

...(5)
From (1), (4) and (5), we get,

straight y space equals space 1 over 15 left parenthesis straight y subscript 2 plus 2 straight y subscript 1 right parenthesis

1 over 10 left parenthesis straight y subscript 2 minus 3 straight y subscript 1 right parenthesis

straight y equals 1 over 15 straight y subscript 2 plus 2 over 15 straight y subscript 1 plus 1 over 10 straight y subscript 2 minus 3 over 10 straight y subscript 1

open parentheses 1 over 15 plus 1 over 10 close parentheses space straight y subscript 2 space plus space open parentheses 2 over 15 minus 3 over 10 close parentheses space straight y subscript 1 minus straight y space equals space 0

open parentheses fraction numerator 2 plus 3 over denominator 30 end fraction close parentheses space straight y subscript 2 plus space open parentheses fraction numerator 4 minus 9 over denominator 30 end fraction close parentheses space straight y subscript 1 space minus space straight y space equals space 0

1 over 6 straight y subscript 2 minus 1 over 6 straight y subscript 1 minus straight y space equals space 0

straight y subscript 2 minus straight y subscript 1 minus 6 straight y equals 0 comma space

which is required differential equation.n

93 Views

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

111 Views

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

101 Views