Solving Differential Equations in Terms of Bessel Functions

10 CHAPTER 1. PRELIMINARIES
Remarks 1.3
1. In general there exists a ∈ K with a ′ = 0 and K[∂] is not commutative.
2. The ring K[∂] is an euclidean ring. For two operators L1,L2 ∈ K[∂],L2 = 0
there are unique operators Q,R ∈ K[∂] such that L1 = QL2 +R and degR < degL2.
This operation is called left division. If R = 0, then Q is called left divisor of L1.
Similarly there exists a right division on K[∂].
An euclidean ring is also a principal ideal ring. Hence, we can define the least
common left multiple LCLM(L1,L2) as the unique monic generator of K[∂]L1 ∩
K[∂]L2 and the greatest common right divisor GCRD(L1,L2) as the unique monic
generator of K[∂]L1 + K[∂]L2. When working with right ideals we can similarly
define the least common right multiple and the greatest common left divisor.
Note that every differential operator L corresponds to a homogeneous differential
equation Ly = 0 and vice versa. We will always assume that L = 0.
Example 1.4
In Maple we can compute the corresponding equation with the following commands:
> with(DEtools):
> L:=Dˆ2-x:
> eq:=diffop2de(L,[D,x],y(x));
eq := −xy(x) + d2
y(x)
dx2 The second parameter [D,x] introduces the variable x and the variable D used
for the derivation ∂. We can define these variables globally:
> Envdiffopdomain:=[D,x]:
Then this parameter can always be omitted, e.g. when computing the corresponding
operator:
> de2diffop(eq,y(x));
∂ 2 − x
From now on we will always assume that the DEtools package is loaded and
that the differential domain [D,x] = [∂,x] is defined.
Definition 1.5 By the solutions of L we mean the solutions of the homogeneous
linear differential equation Ly = 0. They are denoted by V (L).
When talking about differential equations, the term order is commonly used
for the degree of the corresponding operator.
Considering the solutions of Ly = 0 for L ∈ K[∂] a constant factor in K does not
change the solution space V (L). Thus, we can always work with monic operators.