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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)

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)

94 Views

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.

The given equation is
y = e^x (a cos 2x + b sin 2x) ... (1)
Differentiating both sides w.r.t x, we get,
$dy over dx space equals space straight e to the power of straight x left parenthesis straight a space cos space 2 straight x space plus space straight b space sin space 2 straight x right parenthesis space plus space straight e to the power of straight x left parenthesis negative 2 straight a space sin space 2 straight x space plus space 2 straight b space cos space 2 straight x right parenthesis$
$therefore space space space space dy over dx space equals space straight y plus straight e to the power of straight x left parenthesis negative 2 straight a space sin space 2 straight x space plus space 2 straight b space cos space 2 straight x right parenthesis$ ...(2)
$open square brackets because space of space left parenthesis 1 right parenthesis close square brackets$
Again differentiating w.r.t x,
$space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals dy over dx plus straight e to the power of straight x left parenthesis negative 2 straight a space sin space 2 straight x space plus space 2 straight b space cos space 2 straight x right parenthesis space plus space straight e to the power of straight x left parenthesis negative 4 straight a space cos space 2 straight x space minus space 4 straight b space sin space 2 straight x right parenthesis$

or $fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space dy over dx plus open parentheses dy over dx minus straight y close parentheses minus 4 straight e to the power of straight x left parenthesis straight a space cos space 2 straight x space plus space straight b space sin space 2 straight x right parenthesis space space space space space space open square brackets because space of space left parenthesis 2 right parenthesis close square brackets$

or $fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space dy over dx plus dy over dx minus straight y minus 4 straight y space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets$

or $fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 dy over dx plus 5 straight y space equals space 0$
which is required differential equation.