Differential commutator expression stuck

[binbagsss]

Homework Statement

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.

Homework Equations

[/B]
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] ##*

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)## doesn't give me the correct answer.

Thanks in advance.

 

[vela]

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)##.