# Differential commutator expression stuck

• binbagsss
In summary, we are 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. Using the known results ##[D,u]=u_{x}##, ##[D^{2},u]=u_{xx}+2u_{x}D## and the property ##[A,BC]=[A,B]C+B[A,C] ##, we can simplify the expression to ##[u(x),D^{3}]=-[D^{2},u(x)]D-D^{2
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.

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)##.

binbagsss

## 1. What is a differential commutator expression stuck?

A differential commutator expression stuck refers to a mathematical expression that involves differential operators and commutators, which are operators that measure the non-commutativity of other operators. When this expression cannot be simplified or solved, it is considered "stuck".

## 2. How is a differential commutator expression stuck different from a regular mathematical expression?

A differential commutator expression stuck is different because it involves operators and non-commutativity, whereas regular mathematical expressions typically involve only numbers and variables. This makes it more complex and difficult to solve.

## 3. What are some common applications of differential commutator expression stuck in science?

Differential commutator expression stuck is commonly used in quantum mechanics, which studies the behavior of particles at a subatomic level. It is also used in differential geometry, which involves the study of curved spaces, and in other areas of physics and mathematics that deal with non-commutative operators.

## 4. How can one attempt to solve a differential commutator expression stuck?

There are various methods that can be used to solve a differential commutator expression stuck, including using specific formulas, manipulating the expression algebraically, or using numerical methods such as approximation. The best approach may depend on the specific expression and its context.

## 5. What are some challenges that may arise when dealing with a differential commutator expression stuck?

One challenge is that the expression may become increasingly complex as more operators are added, making it difficult to solve or even understand. Additionally, there may not be a clear solution or method for solving the expression, requiring creative problem-solving and advanced mathematical knowledge.

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