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Finding an Operator (from a textbook)

  1. Oct 8, 2014 #1
    1. The problem statement, all variables and given/known data
    Because I wanted to practice more of operators, I borrowed a textbook from a library for extra problems...I managed to solve (a) to (e), but not the last question...which is:

    Write out the operator A2 for A: (f) d2/dx2 - 2xd/dx + 1 for which I keep getting a different solution from the back of the book...which is A2 = d
    4/dx4 - 4xd3/dx3 + (4x2-2)d2/dx2 + 1

    2. Relevant equations
    (given above)

    3. The attempt at a solution
    What I did was:

    Af(x) = d2f(x)/dx2 - 2xdf(x)/dx + f(x)
    A2f(x) = d2/dx2{d2f(x)/dx2 - 2xdf(x)/dx + f(x)} - 2xd/dx{d2f(x)/dx2 - 2xdf(x)/dx + f(x)} + {d2f(x)/dx2 - 2xdf(x)/dx + f(x)}
    = d4f(x)/dx4 - d2/dx2{2xdf(x)/dx} + d2f(x)/dx2 - 2xd3f(x)/dx3 + 4x2d2f(x)/dx2 - 2xdf(x)/dx + d2f(x)/dx2 - 2xdf(x)/dx + f(x)

    ...for d2/dx2(2xdf(x)/dx)...since d2(2x)/dx2 = 0 and d2(df(x)/dx)/dx2 = d3f(x)/dx3

    = d4f(x)/dx4 - 4xd3f(x)/dx3 + (4x2 + 2)d2f(x)/dx2 - 4xdf(x)/dx + f(x)....which is same with the solution ONLY in the first, second and last ones....I still have no idea where I made mistake!
     
  2. jcsd
  3. Oct 9, 2014 #2

    Orodruin

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    Note that for ##\frac{d^2}{dx^2}(x g'(x))##, you will also have terms proportional to ##g''##. These arise when one of the differential operators act on ##x## and the other on ##g'(x)##. Thus, in general,
    $$
    \frac{d^2}{dx^2}\left[x g'(x)\right] \neq x g'''(x).
    $$
     
  4. Oct 9, 2014 #3
    Hello, so....
    d2/dx2(xg'(x)) = g'(x)d2(x)/dx2 + x d2g'(x)/dx2
    =g'(x) d2x/dx2 + x d3x/dx3 ? Is this the right one?

    ...but doesn't d2(x)/dx2 still result in 0 value?
     
  5. Oct 9, 2014 #4

    Orodruin

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    No, I suggest you use one of the differential operators at a time and see what you get instead of guessing.
     
  6. Oct 9, 2014 #5
    So, by "differential operator" do you mean trying to solve each part of A2 separately?

    Just for quick clarification, isn't d/dx {f(x)g(x)} = g(x)df(x)/dx + f(x)dg(x)/dx...doesn't this arrangement apply to d2/dx2 too? Or, do I have to differentiate twice for each f(x) and g(x), in the similar case too, as in d2/dx2 {f(x)g(x)}? I am sorry if I am asking too much but I am trying to re-organize my brain.
     
  7. Oct 9, 2014 #6

    Orodruin

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    I mean that you use the fact that d^2/dx^2 = (d/dx)(d/dx) and apply them one at a time. You will find that the relation you quote is not true for second derivatives.
     
  8. Oct 9, 2014 #7
    Okay, thanks! I will try solving d/dx(d/dx) one at a time!
     
  9. Oct 9, 2014 #8
    I got the solution---thank you!
     
    Last edited: Oct 9, 2014
  10. Oct 9, 2014 #9

    RUber

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    ## A = \frac {d^2}{dx^2} - 2x \frac{d}{dx} + 1 ## so A^2 is A acting on itself. I don't think you need to include f(x) yet. However, leaving it out means you need to think of (1) as the identity.
    ##A^2=(\frac {d^2}{dx^2} - 2x \frac{d}{dx} + 1)(\frac {d^2}{dx^2} - 2x \frac{d}{dx} + 1)##
    ##A^2=\frac {d^2}{dx^2}\frac {d^2}{dx^2} - \frac {d^2}{dx^2}2x \frac{d}{dx} + \frac {d^2}{dx^2}(1)\\
    - 2x \frac{d}{dx}\frac {d^2}{dx^2} +2x \frac{d}{dx}2x \frac{d}{dx} - 2x \frac{d}{dx}(1)\\
    +\frac {d^2}{dx^2} - 2x \frac{d}{dx} + 1##
    ##A^2=\frac {d^4}{dx^4} - \frac {d^2}{dx^2}2x \frac{d}{dx}+ \frac {d^2}{dx^2} - 2x \frac {d^3}{dx^3} +2x \frac{d}{dx}2x \frac{d}{dx} - 2x \frac{d}{dx}+\frac {d^2}{dx^2} - 2x \frac{d}{dx} + 1##
    It seems like the most likely location for an error would be in the ##\frac {d^2}{dx^2}2x \frac{d}{dx}## and ## 2x \frac{d}{dx}2x \frac{d}{dx}## terms.
    It looks like your expansion above is right for ##\frac{d}{dx}2x\frac{d}{dx}=2\frac{d}{dx}+2x\frac{d^2}{dx^2}## and the derivative of that you have above looks right as well.
    The ##-4x\frac{d}{dx}## term looks like you forgot to expand out the ##+2x\frac{d}{dx}2x\frac{d}{dx} ## term fully to produce a ##+4x\frac{d}{dx}## to cancel out the ##-4x\frac{d}{dx}##.
     
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