Linear Differential Operator order

[tom_rylex]

[Solved] Linear Differential Operator order

Homework Statement

I'm misunderstanding something basic about how this works:

Give examples of linear differential operators $$L$$ and $$M$$ for which it is not true that $$L(M(u)) = M(L(u))$$ for all u.

Homework Equations

Since it's arbitrary, I made two first order differential functions of two variables:
$$\overline{x} = {x,y}$$
$$u=u(x,y)$$
$$L(u) = a(\overline{x})u + b_1(\overline{x})u_x + b_2(\overline{x})u_y$$
$$M(u) = c(\overline{x})u + d_1(\overline{x})u_x + d_2(\overline{x})u_y$$
Where a,b,c, and d are coefficients.

The Attempt at a Solution

When I expand $$L(M(u))$$, it seems to look like $$M(L(u))$$:

$$L(M(u)) = a(\overline{x})c(\overline{x})u+a(\overline{x})d_1(\overline{x})u_x +a(\overline{x})d_2(\overline{x})u_y + b_1(\overline{x})c(\overline{x})u_x+b_2(\overline{x})c(\overline{x})u_y$$

$$M(L(u)) = c(\overline{x})a(\overline{x})u+c(\overline{x})b_1(\overline{x})u_x +c(\overline{x})b_2(\overline{x})u_y + d_1(\overline{x})a(\overline{x})u_x+d_2(\overline{x})a(\overline{x})u_y$$

I'm stuck, since it seems that all roads lead back to a commutative relationship.

 

[HallsofIvy]

Linear differential operators with constant coefficients are commutative.

Try using linear differential operators with variable coefficients. That is, something like (d/dx+ 3x)y or (xd/dx)u

 

[tom_rylex]

[Solved] Linear Differential Operator order

Thanks. I just needed a nudge to get in the right direction. Some parts of DiffEq have been a while for me. I set up the linear differential operators

$$L = \frac {d}{dx}$$
$$M = \frac {d}{dx} + x$$

to show that $$L(M(u)) \neq M(L(u))$$ for all u. I think you helped me become 2 points smarter (don't ask me the scale...).