Lecture Notes in Differential Equations - Bruce E. Shapiro

276 LESSON 29. REGULAR SINGULARITIES
Proof. This is a restatement of theorem 28.1.
Example 29.1. The differential equation (1 + t 2 )y ′′ + 2ty ′ + 4t 2 y = 0 has
singularities at t = ±i. The series solution ∑ a k t k about t 0 = 0 has a radius
of convergence of 1 while the series solution ∑ b k (t − 1) k about t 0 = 1 has
a radius of convergence of √ 2.
The Cauchy-Euler equation
t 2 y ′′ + αty ′ + βy = 0 (29.4)
where α, β ∈ R is the canonical (standard or typical) example of a differential
equation with a regular singularity at the origin. It is useful because
it provides us with an insight into why the Frobenius method that we will
discuss in chapter 30 will work, and the forms and methods of solution
resemble (though in simpler form) the more difficult Frobenius solutions to
follow.
Example 29.2. Prove that the Cauchy-Euler equation has a regular singularity
at t = 0.
Comparing with equation 29.1, we have that a(t) = t 2 , b(t) = αt, and
c(t) = β. First, we observe that a(0) = 0, hence there is a singularity at
t = 0. Then, applying the definition of a regular singularity (definition
29.1) with t = 0,
(t) · αt
lim
t→0 t 2 = α (29.5)
lim
t→0
(t 2 ) · β
t 2 = β (29.6)
Since both α and β are given real number, the limits exist and are finite.
Hence by the definition of a regular singularity, there is a regular singularity
at t = 0.
The following example demonstrates one of the problem that arises when we
go blindly ahead and attempt to find a series solution to the Cauchy-Euler
equation.
Example 29.3. Attempt to find a series solution about t = 0 to the
Cauchy-Euler equation (29.4).
We begin by letting y = ∑ ∞
k=0 a kt k in (29.4), which gives