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Math Help - Cauchy-Euler Dif EQ

  1. #1
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    Cauchy-Euler Dif EQ

    For a Cauchy Euler Dif EQ, its auxiliary equation for this Dif EQ:

    at^2y'' + bty' + cy = g(t) is:

    am^2 + (b-a)m + c = 0.

    Use this auxiliary equation to solve the Cauchy-Euler equation below:

    t^2y'' - 5ty' + 8y = 0 subject to y(2) = 32 and y'(2) = 0
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by alikation0 View Post
    For a Cauchy Euler Dif EQ, its auxiliary equation for this Dif EQ:

    at^2y'' + bty' + cy = g(t) is:

    am^2 + (b-a)m + c = 0.

    Use this auxiliary equation to solve the Cauchy-Euler equation below:

    t^2y'' - 5ty' + 8y = 0 subject to y(2) = 32 and y'(2) = 0
    Well, a = 1, b = -5 and c = 8, so the auxiliary equation becomes
    m^2 - 6m + 8 = 0

    (m - 2)(m - 4) = 0

    So m = 2 and m = 4.

    Thus the most general solution to this equation is
    y = Ax^2 + Bx^4

    -Dan
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  3. #3
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    If you want a solution on (0,\infty) then it is like topsquark said. If you want a solution on the open set (-\infty,0)\cup (0,\infty) then it is A|x|^2+B|x|^4.
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