Finding a linear differential operators to a problem I need to solve the following problem: 

Some reference:  

Any ideas where to start? I get confused when I try to solve this myself: 
$$M(L(u)) = M(a(\boldsymbol{x})u + \sum_{i=1}^n b_i(\boldsymbol{x})\frac{\partial u}{\partial x_i}+\cdots) = M(a(\boldsymbol{x})u) + M(\sum_{i=1}^n b_i(\boldsymbol{x})\frac{\partial u}{\partial x_i}) + \cdots$$
What does for example $M(a(\boldsymbol{x})u)$ evaluate into? The linear differential operator is applied into the product of two functions of $\boldsymbol{x}$, yet I see no definition for this in the above text? 
 A: It is very easy to construct such examples within the framework of first-order linear differential operators $X$ with vanishing constant term; that is, of the general form
$X = \sum_1^n X_i \dfrac{\partial}{\partial x_i}; \tag{1}$
as an easy instance, set 
$M = \sum_1^n M_i \dfrac{\partial}{\partial x_i}, \tag{2}$
where the $M_i$ are constant, and 
$L = \sum_1^n x_j \dfrac{\partial}{\partial x_j}. \tag{3}$
Then for any sufficiently differentiable (i.e., twice or of class $C^2$) function $u$, 
$L(M(u)) = \sum_{i, j = 1}^{i, j = n} M_ix_j \dfrac{\partial^2u}{\partial x_i \partial x_j}, \tag{4}$
whereas
$M(L(u)) =  \sum_{i, j = 1}^{i, j = n} M_ix_j \dfrac{\partial^2u}{\partial x_i \partial x_j} + \sum_1^n M_i \dfrac{\partial u}{\partial x_i}, \tag{5}$
so that in fact
$L(M(u)) - M(L(u)) = -M(u). \tag{6}$
It is very easy to verify (4)-(6) by simple calculations, so I leave them for my readers.  The above gives a simple example of a broad class of operators which do not commute when applied to such functions $u$.  Allowing $L$ and $M$ to have more complex functions as coefficients provides many more instances of this phenomenon.
What does $M(a(\boldsymbol{x})u)$ evaluate to?  It gets more and more complex as the order of $M$ grows, but for the case of $M$ as above it is simply $M(a(\boldsymbol{x}))u + a(\boldsymbol{x})M(u)$, an easy consequence of the Leibniz rule for the (first-order) derivatives of products.  And it's too close to my bedtime for me to start $\LaTeX$'ing up the more complicated formulas which arise when expressions with more complex coefficients and/or of higher order are considered.  But I think some good leads on the general picture are given here.  A lot of algebraic grinding is necessary to work these things out for higher order cases with more complicated coefficients, but that is about all.
For a much more thorough discussion of the properties of these first-order operators, in particular of their commutativity properties, see my answer to this question.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
