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Homework Help: Why doesn't this method work? (Re: Simultaneous ODEs)

  1. May 10, 2010 #1
    I have been working on a derivation in which the following simultateous ordinary differential equations have appeared:
    [tex] f^{(4)}(x)-2 a^2 f''(x)+a^4 f(x)+b(g''(x)-a^2 g(x))=0,[/tex]
    [tex] g^{(4)}(x)-2 a^2 g''(x)+a^4 g(x)-b(f''(x)-a^2 f(x))=0,[/tex]
    where [tex]a[/tex] and [tex]b[/tex] are constants. I figured that I could solve this using Fourier transforms. First, I transform the above, using [tex]\nu[/tex] as the transform space analogue of [tex]x[/tex], into the following:
    [tex] (\nu^2 +a^2 )\hat f(\nu ) -b \hat g(\nu )=0,[/tex]
    [tex] (\nu^2 +a^2 )\hat g(\nu ) +b \hat f(\nu )=0.[/tex]
    I then rearrange these equations via substitution into
    [tex] [(\nu^2 +a^2 )^2 +b^2 ] \hat f(\nu ) =0,[/tex]
    [tex] [(\nu^2 +a^2 )^2 +b^2 ] \hat g(\nu )=0.[/tex]
    Taking the inverse Fourier transform then results in
    [tex] f^{(4)}(x)-2a^2 f''(x)+(a^4 +b^2 )f(x)=0,[/tex]
    [tex] g^{(4)}(x)-2a^2 g''(x)+(a^4 +b^2 )g(x)=0.[/tex]

    However, after I solve for these two ODEs using the boundary conditions, I find that the resulting answer is erroneous. I can only conclude that the above methodoly doesn't work, although I can't see why.

    Could anyone point out the mathematical error in the above steps?
     
    Last edited: May 10, 2010
  2. jcsd
  3. May 10, 2010 #2

    lanedance

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    can you show your transform... shouldn't you get factors of [itex]nu^4[/itex]?
     
    Last edited: May 10, 2010
  4. May 10, 2010 #3

    vela

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    The OP is dividing out a common factor of [itex]\nu^2+a^2[/itex].
     
  5. May 13, 2010 #4
    Yes, that's what I've done.
     
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