Lecture Notes in Differential Equations - Bruce E. Shapiro

294 LESSON 30. THE METHOD OF FROBENIUS
By Taylor’s theorem p (k) (t 0 ) = k!p k and S (n−k) (t 0 ) = (n − k)!c n−k , where
p k and c k are the Taylor coefficients of p(t) and S(t), respectively,
Similarly, the third term in (30.97) is
D (n) {[α(α − 1) + αp(t) + q(t)] S} (t 0 )
n∑
( ) n {
= [α(α − 1) + αp(t) + q(t)] (k) (t
k
0 )}
S (n−k) (t 0 ) (30.109)
k=0
Extracting the first (k = 0) term,
D (n) {[α(α − 1) + αp(t) + q(t)] S} (t 0 )
= [α(α − 1) + αp 0 + q 0 ] S (n) (t 0 )
n∑
( ) n [
+ αp (k) (t
k
0 ) + q (k) (t 0 )]
S (n−k) (t 0 ) (30.110)
k=1
By the indicial equation, the first term is zero. Substituting the formulas
for the Taylor coefficients and simplifying,
n∑
(
D (n) n
{[α(α − 1) + αp(t) + q(t)] S} (t 0 ) = [αk!p
k)
k + k!q k ] (n−k)!c n−k
k=1
Substituting (30.111) and (30.102) into (30.97),
(30.111)
0 =n(n − 1)c n n! + nn!(2α + p 0 )c n
n−1
∑
n∑
+ n!(n − k)p k c n−k + n! [αp k + q k ] c n−k (30.112)
k=1
Rearranging and simplifying,
k=1
n−1
∑
n(n − 1 + 2α + p 0 )c n + [(n − k + α)p k + q k ]c n−k = 0 (30.113)
k=1
Letting r > 0 be the radius of convergence of the Taylor series for p and
q, then there is some number M such that |p k | r k ≤ M and |q k | r k ≤ M.
Then
|n(n − 1 + 2α + p 0 )c n | ≤ M
n−1
∑
k=1
1
r k [|n − k + α| + 1] |c n−k| (30.114)
Define the numbers C 0 = |c 0 | , C 1 = |c 1 |, and, for n ≥ 2,
|n − 1 + 2α + p 0 | C n = M n
n−1
∑
k=1
1
r k [|n − k + α| + 1] C n−k (30.115)