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Stuck on the reduction of order step for solving this differential equation

  1. Mar 5, 2012 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Find the general solution for the equation

    (x - 1)y'' - xy' + y = sin(x), x > 1

    Given that y_1(x) = e^x satisfies the associated homogeneous equation.


    2. Relevant equations
    y_2 = v_2(x) * y_1


    3. The attempt at a solution
    I read http://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx and attempted to replicate its method several times and I am attaching my latest attempt. The website I linked to says "Note that upon simplifying the only terms remaining are those involving the derivatives of v. The term involving v drops out. If you’ve done all of your work correctly this should always happen." but I have a term involving v that did not drop out. Also, am I supposed to ignore sin(x) or not? Based on the way the question is phrased, I'd now say I should of ignored it (please tell me if I am correct in saying this) but it doesn't matter for what I am questioning.

    By the way, this thread is just about the reduction of order part. (The next step is variation of parameters but I haven't gotten there yet.)

    Any help would be greatly appreciated!
    Thanks in advance!
     

    Attached Files:

  2. jcsd
  3. Mar 5, 2012 #2

    SammyS

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    Check your algebra.

    There is no term left involving v.
     
  4. Mar 6, 2012 #3

    s3a

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    I found that the ve^x is supposed to be vxe^x such that they do cancel out (thanks) but now I'm stuck again. Could you please tell me what I am doing wrong now?

    If I'm right so far, I don't see how what I did yields y = x.
     

    Attached Files:

  5. Mar 6, 2012 #4

    ehild

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    You got v' correctly,

    [tex]v'=C_1e^{-x}(x-1)[/tex].

    Integrate, add second constant, multiply by ex.

    ehild
     
  6. Apr 20, 2012 #5

    s3a

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    Sorry for the late reply but thanks :).
     
  7. Apr 21, 2012 #6

    ehild

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    Better late than never:wink:

    ehild
     
  8. Apr 22, 2012 #7

    s3a

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    Lol ya. :D
     
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