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Variation of parameter method

  1. Nov 28, 2009 #1
    1. The problem statement, all variables and given/known data

    (D^2 + 2D + 1)y = ln(x)/(xe^x)

    2. Relevant equations

    D = d/dx

    3. The attempt at a solution

    First I find the roots of the left side of the equation, -1 of multiplicity 2.
    This leads to
    y(c) = Ae^(-x) + Bxe^(-x)

    Substituting A and B with a' and b' and dividing both sides by e^(-x) I find the two equations:

    -a' + (1-x)b' = ln(x)/x
    a' + xb' = 0

    Which leads to a' = -ln(x) and b' = ln(x)/x

    Integrating by parts leads to:

    a = x(1-ln(x))
    b = (1/2)[ln(x)]^2

    Which leads to

    y(p) = ae^(-x) + bxe^(-x)
    = xe^(-x)[1-ln(x) + (1/2)(ln(x))^2]

    So y = xe^(-x)[1 - ln(x) + (1/2)(ln(x))^2 + A/x + B]

    Does this seem correct?
     
  2. jcsd
  3. Nov 28, 2009 #2

    Dick

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    It matches what I get. You might notice that the '1' doesn't need to be there, you could just merge it into the constant B.
     
  4. Nov 28, 2009 #3
    Thanks a bunch for the reply. I am new to these forums and wondering if there is a "Thank" button or anything like I've seen on other similar forums.
     
  5. Nov 28, 2009 #4

    Dick

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    Not that I know of. But thanks for looking for it.
     
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