Solving Differential Equations in Terms of Bessel Functions

34 CHAPTER 2. TRANSFORMATIONS
they exist. If we found those transformations, we also found V (L). Note that we
also need to find the parameter ν of the Bessel functions involved.
We know from the previous theorem that if those transformations exist, there
exists M ∈ K[∂] such that
LB −→C M −→EG L.
We will address those two parts separately.
But first we will clarify the next steps using some examples.
Example 2.11
1. We consider the modified Bessel operator LB with ν = 2:
L = x 2 ∂ 2 + x∂ − x 2 + 4 .
Using the results of Example 1.32 we know that the generalized exponents at x = 0
and x = ∞ are
gexp(L,0) = {2,−2} (2.6)
and gexp(L,∞) =
Now we apply a change of variables
to L:
x → f (x) =
1
T
> L:=xˆ2*Dˆ2+x*D-(xˆ2+2ˆ2):
1 1 1
+ ,− +
2 T 2
2(x − 1)(x − 2)2
(x − 3) 2
> f:=2*(x-1)*(x-2)ˆ2/(x-3)ˆ2:
> M:=changeOfVars(L,f);
M :=(x − 2) 3 x 2 − 7x + 8 (x − 3) 6 (x − 1) 3 ∂ 2 +
4 3 2 5 2 2
x − 14x + 55x − 84x + 46 (x − 3) (x − 1) (x − 2) ∂−
4 x 2 3
− 7x + 8 x 6 − 10x 5 + 42x 4 − 100x 3 + 158x 2
− 172x + 97
(x − 2)(x − 1)
We will now assume that M is given. We want to find the parameter of the
change of variables that sends L to M, i.e. we want to find f using only the operator
M.
.