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Homework Help: Differential Operators

  1. Sep 3, 2005 #1
    We're doing differential operators in my Differential Equations class right now, and our professor assigned the following problem to us:

    (D-x)(D+x)

    Which inevitably gives us the following terms as part of the final answer: Dx-xD

    The answer in the book tells me that Dx-xD = 1, and some preliminary research has told me that this is true. What I couldn't find was the why. Why, or how, does Dx-xD result in 1? And does -Dx+xD = -1?
     
  2. jcsd
  3. Sep 3, 2005 #2

    lurflurf

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    Homework Helper

    product rule gives
    Dx=1+xD
    might be easier to see with a function
    D(xy)=(Dx)y+x(Dy)=y+x(Dy)=(1+xD)y
    so
    Dx=1+xD
     
  4. Sep 3, 2005 #3
    That is because
    [tex]\frac{d}{dx}\left(xf(x)\right)-x\frac{d}{dx}\left(f(x)\right)=xf'(x)+f(x)-xf'(x)=f(x)[/tex]
    [tex](Dx-xD)f(x)=f(x)[/tex]
    Hence Dx-xD=1
     
  5. Sep 3, 2005 #4
    Thanks! o:)
     
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