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Differential Equations

Short Answer Type

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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Long Answer Type

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

Let the radius of circle = a
Since the circle has centre on x-axis and passes through the origin.
∴ its centre is (a, 0)
The equation of circle is
( x – a)²+( y – 0)² = a² or x² –2 a x + a² + y² = a²∴ x²+ y² – 2 ax = 0 ...(1)
Differentiating both sides w.r.t. x, we get.
$2 straight x plus 2 straight y dy over dx minus 2 straight a space equals space 0 space space space space space space space space space space space space or space space space space space space straight x plus straight y dy over dx minus straight a space equals space 0$
$therefore space space space space space space space space space space space straight a space equals space straight x plus straight y dy over dx$
Putting this value of a in (1), we get.
$straight x squared plus straight y squared minus 2 straight x open parentheses straight x plus straight y dy over dx close parentheses space equals space 0$
$therefore space space space straight x squared plus straight y squared minus 2 straight x squared minus 2 xy dy over dx space equals space 0 therefore space space space space minus straight x squared plus straight y squared minus 2 xy dy over dx space equals space 0 therefore space space space space space space 2 space xy dy over dx plus straight x squared minus straight y squared space equals space 0$
which is required differential equation.

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Short Answer Type

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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