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Differential commutator expression stuck

  1. May 26, 2015 #1
    1. The problem statement, all variables and given/known data

    I am trying to show that ##a(x)[u(x),D^{3}]=-au_{xxx}-3au_{xx}D-3au_{x}D^{2}##, where ##D=d/dx##, ##D^{2}=d^{2}/dx^{2} ## etc.


    2. Relevant equations

    I have the known results :

    ##[D,u]=u_{x}##
    ##[D^{2},u]=u_{xx}+2u_{x}D##

    The property: ##[A,BC]=[A,B]C+B[A,C] ##*


    3. The attempt at a solution

    Let me drop the ##a(x)## and consider ##[u(x),D^{3}]##

    ##=[u(x),D^{2}(D)] = [u(x),D^{2}]D+D^{2}[u(x),D]## using *

    ##=-[D^{2},u(x)]D-D^{2}[D,u(x)]=-u_{xx}D-2u_{x}D-D^{2}u_{x}=-u_{xx}D-2u_{x}D-u_{xxx}##, using the 2 results quoted above.

    Multiplying by ##a(x)## doesnt give me the correct answer.

    Thanks in advance.
     
  2. jcsd
  3. May 26, 2015 #2

    vela

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    I suggest you use parentheses and take things one step at a time. You seem to have the basic idea, but you're messing up the algebra. I'd also write ##D^3## as ##D(D^2)##. That way the ##D^2## ends up to the right of the commutator, so you don't have to differentiate a product twice, i.e., ##D^2[u,D]## isn't just ##D^2 u_x = u_{xxx}## because ##D^2[u,D]f = D^2(u_x f)##.
     
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