Lecture Notes in Differential Equations - Bruce E. Shapiro

297
is a solution of
(t − t 0 ) 2 y ′′ + (t − t 0 )p(t)y ′ + q(t)y = 0 (30.134)
with radius of convergence r (as was shown in theorem 30.1), and a second
linearly independent solution, also with radius of convergence r, is given by
one of the following three cases.
1. If α 1 = α 2 = α, then
2. If ∆ ∈ Z, then
y 2 = ay 1 (t) ln |t − t 0 | + (t − t 0 ) α
y 2 = ay 1 (t) ln |t − t 0 | + (t − t 0 ) α2
∞ ∑
k=0
∞
∑
k=0
a k (t − t 0 ) k (30.135)
a k (t − t 0 ) k (30.136)
3. If ∆ ∉ Z, then
y 2 = (t − t 0 ) α2
∞ ∑
k=0
a k (t − t 0 ) k (30.137)
for some set of constants {a, a 0 , a 1 , ...}.
Proof. We only give the proof for t 0 = 0; otherwise, make the change of
variables to x = t − t 0 and the proof is identical. Then the differential
equation becomes
t 2 y ′′ + tp(t)y ′ + q(t)y = 0 (30.138)
and the first Frobenius solution (30.133) is
u(t) = t α
∞ ∑
k=0
c k t k (30.139)
where α is the larger of the two roots of the indicial equation. By Abel’s
formula the Wronskian of (30.138)
{ ∫ } {
p(t)
∞∑
∫ }
W (t) = exp − dt = exp − p k t k−1 dt (30.140)
t
where {p 0 , p 1 , ...} are the Taylor coefficients of p(t). Integrating term by
term
{
}
{ }
∞∑ p k
∞∑ p
W (t) = exp −p 0 ln |t| −
k tk = |t| −p0 k
exp −
k tk (30.141)
k=1
k=0
k=1