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Operator Equation

  1. Sep 21, 2007 #1
    1. The problem statement, all variables and given/known data

    Verify the operator equation

    (d/dx + x)(d/dx - x) = d^2/dx^2 - x^2 -1

    (where d is meant to be the partial derivative symbol)

    2. Relevant equations

    None that are obvious to me?

    3. The attempt at a solution

    The truth is i'm not really sure how i should be going about this i tried expanding the brackets to get:

    d^2/dx^2 + d/dx*x - d/dx*x - x^2

    I didn't know whether having d/dx*x means i should differentiate or just leave it but i thought they'd cancel either way giving d^2/dx^2 - x^2 so where does the -1 come from?
     
  2. jcsd
  3. Sep 21, 2007 #2
    You can't reorder things:

    [tex]\left(\frac{d}{dx} + x\right)\left(\frac{d}{dx} - x\right) = \frac{d^2}{dx^2} + x \frac{d}{dx} - \frac{d}{dx}x - x^2[/tex]
     
  4. Sep 21, 2007 #3

    Dick

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    Let your operator operate on something, like f(x). That will make what's going on much clearer.
     
  5. Sep 21, 2007 #4
    I think i understand why it's important to keep the order so if i applied the operator to f(x) i think it'd become:

    d^2/dx^2(f[x]) + x*d/dx(f[x]) - d/dx(x*f[x]) - x^2 *f[x]

    f''(x) + x*f'(x) - {f(x) + x*f'(x)} - x^2 *f(x)

    f''(x) + x*f'(x) - x*f'(x) - x^2 *f(x) - f(x)

    f''(x) - x^2 *f(x) - f(x)

    which is the same as if i'd done this to the function? d^2/dx^2 - x^2 -1
     
  6. Sep 21, 2007 #5

    Dick

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    I think you are right.
     
  7. Sep 21, 2007 #6
    Thanks for the help :)
     
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