So with a notation capital D
is equal to d over dx and the family of constant a_sub_i,
we call any expression of
the form P of capital D which is equal to a_sub_n times D to the n,
plus a_sub_(n-1) times D to the n-1,
plus a_sub_1 times D plus a_sub_0 are differential polynomial.
Any such expression with
the constant coefficient is a differential polynomial.
Then it's easy to see
that any two differential polynomials P(D) and the Q(D), they commute.
In other words, P(D) times Q(D)
acting on y is equal to Q(D) times P(D) acting on y.
We can switch the order of operation to differential operators P(D) and Q(D).
We can exchange them.
From P(D)Q(D) into Q(D)P(D).
As a very simple example,
let's consider the (D-2) times (D+1) of y.
What is it? (D-2) times and the (D+1) of y,
forced acting (D+1) on y,
then you will get (D-2),
D y means y prime and plus one times y that is y.
Apply (D-2) again, then apply D on this expression.
You will get y double prime plus
y prime minus 2 times of this is minus 2 y prime, minus 2 y.
So, finally, we'll get y double prime minus y prime minus 2 y,
which is the same as this D squared minus D minus 2 times of y.
On the other hand,
if you exchange the order of operations for these two differential polynomials,
then you will get d plus one times d minus two.
On the other hand,
(D+1) times (D-2) of y which is equal to
(D+1) and D acting on y is the y prime and minus two times of y,
that is 2 y.
Now apply D again,
then you will get y double prime minus 2
y prime plus one times of this will be y prime minus 2 y,
which is equal to y double prime minus y prime minus 2 y,
which is exactly equal to this expression.
So that we confirm that these two.
(D-2) times (D+1) is the same as (D+1) times (D-2).
In fact, they are equal to this D squared - D - 2.
But be careful, such a commutativity then not be true in general.
For example, if the differential operator has
a variable coefficient then you
cannot exchange the order of those two differential operators.
As a very simple example,
consider the x D and D acting on y,
you will get x of D and y prime that is equal to x y double prime.
However, if you exchanged the order of these two,
then you will get D times x D of y.
So that is equal to D of x y prime.
Then, this is equal to the by the product rule,
x y double prime and plus y prime.
So that these two things are not the same thing.
In other words, you cannot exchange the order of these two operations,
x D times D is not equal to D times x of D.
It's because the one of the operator x D has variable coefficient.