Lecture Notes in Differential Equations - Bruce E. Shapiro

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Example 30.4. Find a Frobenius solution to Bessel’s equation of order
1/2 near the origin,
t 2 y ′′ + ty ′ + (t 2 − 1/4)y = 0 (30.41)
By “near the origin” we mean “in a neighborhood that includes the point
t 0 = 0.
The differential equation has the same form as y ′′ + b(t)y ′ + c(t)y = 0 with
b(t) = 1/t and c(t) = (t 2 − 1/4)/t 2 , neither of which is analytic at t = 0.
Thus the origin is not an ordinary point. However, since
and
p(t) = tb(t) = 1 (30.42)
q(t) = t 2 c(t) = t 2 − 1/4 (30.43)
are both analytic at t = 0, we conclude that the singularity is regular.
Letting t 0 = 0, we have p(t 0 ) = p(0) = 1 and q(t 0 ) = −1/4, so the indicial
equation is
0 = α(α − 1) + α − 1 4 = α2 − 1 4
(30.44)
The roots are α = ±1/2, so the two possible Frobenius solutions
y 1 = √ ∞∑
∞∑
t a k t k = a k t k+1/2 (30.45)
k=0
k=0
k=0
y 2 = √ 1 ∑
∞ ∞∑
b k t k = b k t k−1/2 (30.46)
t
Starting with the first solution,
∞∑
y 1 ′ = a k (k + 1/2)t k−1/2 (30.47)
y ′′
1 =
k=0
k=0
∞∑
a k (k 2 − 1/4)t k−3/2 (30.48)
k=0
Using (30.45) and (30.47) the differential equation gives
∞∑
(
0 = a k k 2 − 1 )
∞∑
(
t k+1/2 + a k k + 1 )
t k+1/2
4
2
k=0
k=0
(
+ t 2 − 1 )
∑ ∞
a k t k+1/2 (30.49)
4
k=0