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Homework Help: General solution of ode using fourier transform

  1. Apr 9, 2010 #1
    ok well i'm pretty much home and dry in this problem

    the aim of this problem is to get the general solution for the ode below..

    2u'' - xu' + u = 0 = g(x)
    i started to solve it by rearranging the equation..

    2u'' + u = xu'

    apply fourier transform..
    2F(u'') + u^ = g^

    (-2k^2)u^ + u^ = g^

    u^ [1- 2(k^2)] = g^
    u^ = {1/ [1- 2(k^2)]}g^

    the problem is, i cant find any of the them in the transform table..
     
  2. jcsd
  3. Apr 10, 2010 #2

    vela

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    So g(x)=0?
    Where did the xu' term go, and where did g come from?
     
  4. Apr 10, 2010 #3
    rearranging my equations...

    2u'' + u = xu' = g(x)
    where xu' = g(x)

    F{ 2u'' + u = g(x) }
    -2k^2(u^) + (u^) = (g^)
    (1 - k^2 )(u^) = (g^)
    u^ = [1 / (1 - k^2 )] (g^)

    correct?
     
  5. Apr 10, 2010 #4

    vela

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    Sort of, but it's not what you want to do. What exactly is g(x) supposed to be? According to your first equation (in your original post), g(x)=0. Or is it supposed to be the source term/forcing function, i.e. the term that results in the particular solution?

    Try looking up a property of the Fourier transform relating [itex]xf(x)[/itex] to [tex]\frac{d}{dk}\hat{f}(k)[/tex].
     
  6. Apr 10, 2010 #5
    xu' --->> [- (1/ i2pi)][d/dx]u

    correct?
    should i substitute it to my previous equation?
     
    Last edited: Apr 10, 2010
  7. Apr 10, 2010 #6

    vela

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    Try it out.
     
  8. Apr 10, 2010 #7
    F(u'' -xu' + u = 0)
    F(-xu') = -[(-1 / i2pi)(d/dx)u^])
    F(-xu') = [(1 / i2pi)(d/dx)u^])

    F(u'' -xu' + u = 0)
    [1 -2k^2 + (1 / i2pi)(d/dx)]u^ = 0

    correct?
     
  9. Apr 11, 2010 #8

    vela

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    No, you seem to be missing a factor of k. First, take care of the effect of the x:

    [tex]F[xu']=\frac{1}{-i}\frac{d}{dk}F[u'][/tex]

    And then take the Fourier transform of u':

    [tex]F[xu']=\frac{1}{-i}\frac{d}{dk}(-ikF)=\frac{d}{dk}[k\hat{u}(k)][/tex]

    (If you're using x and k, I think there are no factors of 2π.)
     
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