Chapter 5.2, Problem 39E

Let $A$, $B$, and $C$ represent three linear differential operators with constant coefficients; for example,

$A:=a_2{D}^2+a_{1}D+a_0$, $B:=b_2{D}^2+b_{1}D+b_0$, $C:=c_2{D}^2+c_{1}D+c_0$,

where the $a$’s, $b$’s, and $c$’s are constants. Verify the following properties: ${}^{†}$

${}^{†}$ We say that two operators $A$ and $B$ are equal if $A\left[y\right]=B\left[y\right]$ for all functions $y$ with the necessary derivatives.

a. Commutative laws:

$A+B=B+A$,

$AB=BA$.

b. Associative laws:

$\left(A+B\right)+C=A+\left(B+C\right)$,

$\left(AB\right)C=A\left(BC\right)$.

c. Distributive law: $A\left(B+C\right)=AB+AC$.