Question

Suppose L is a linear differential operator that factors but has variable coefficients. Do the factors ofLcommute? Defend your answer.

Short answer

No, the factor does not commute.

Step-by-step solution

Definition of commutative property

The commutative property states that the numbers we operate can be moved or swapped from their position without making any difference to the answer.

Given data

Consider a differential operator, L that factors into

$L_1,L_2$where$L_1=\left(xD-1\right)$

And$L_2=\left(D+4\right)$

For L1L2x

Now, find$L_1{L}_{2}x$ as follows:

$\begin{array}{rcl}{L}_1{L}_{2}x& =& \left(xD-1\right)\left(D+4\right)x\\ & =& \left(xD-1\right)\left(Dx+4x\right)\\ & =& \left(xD-1\right)\left(1+4x\right)\\ & =& xD\left(1\right)+xD\left(4x\right)-1-4x\\ & =& x\left(0\right)+4x-1-4x\\ & =& -1\end{array}$

For L2L1x

And$L_2{L}_{1}x=\left(D+4\right)\left(xD-1\right)x$

$$\begin{gathered} =\left(D+4\right)\left(xDx-x\right) \\ =\left(D+4\right)\left(x·1-x\right) \\ =\left(D+4\right)\left(0\right) \\ =0 \end{gathered}$$

Thus, the operators do not commute as$L_1{L}_{2}x\ne L_2{L}_{1}x$ .