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Ordinary Differential Equations

  1. Feb 26, 2007 #1
    1. The problem statement, all variables and given/known data
    Use variation of parameters to find the general solution of:

    [tex] x^2 y'' + xy' + 9y = -tan(3ln(x)) [/tex]


    2. Relevant equations

    [tex]y_p = y_2 (x) \int \frac{y_1 (x) g(x)}{W[y_1 (x),y_2 (x)]} dx - y_1 (x) \int \frac{y_2 (x) g(x)}{W[y_1 (x),y_2 (x)]} dx [/tex]

    3. The attempt at a solution

    Solution for the homogeneous Euler equation is given by
    [tex]
    y_1 (x) = cos(3ln(x)), y_2 (x) = sin(3ln(x))
    [/tex]

    I try using these solutions in the equation for variation of parameters with:
    [tex] W[y_1 , y_2] = \frac{3}{x} [/tex]
    [tex] g(x) = -tan(3ln(x)) [/tex]

    The first term I can evaluate, but the second term gives me:
    [tex] y_1 (x) \int \frac{x sin^2 (3ln(x))}{3 cos(3ln(x))} dx [/tex]

    which doesn't seem solvable by any methods I know...

    Any suggestions on what to do from here, or where I went wrong on the way would be helpful.
     
  2. jcsd
  3. Feb 26, 2007 #2
    You have to set it up as y" +..... to use your variation of parameters formula
     
  4. Feb 26, 2007 #3
    so the solutions for the homogeneous equation are the same, right? but I need to divide through by [itex] x^2 [/itex] before I apply the formula? so
    g(x) becomes:

    [tex] g(x) = -\frac{tan(3ln(x))}{x^2} [/tex]
     
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