Lecture Notes in Differential Equations - Bruce E. Shapiro

262 LESSON 28. SERIES SOLUTIONS
Ordinary and Singular Points
The method of power series solutions we have described will work at any ordinary
point of a differential equation. A point t = t 0 is called an ordinary
point of the differential operator
L n y = a n (t)y (n) + a n−1 (t)y (n−1) + · · · + a 0 (t)y (28.61)
if all of the functions
p k (t) = a k(t)
a n (t)
(28.62)
are analytic at t = t 0 , and is called a singular point (or singularity) of
the differential operator if any of the p k (t) are not analytic at t = t 0 . If not
all the p k (t) are analytic at t = t 0 but all of the functions
namely,
q k (t) = (t − t 0 ) n−k p k (t) (28.63)
q n−1 (t) = (t − t 0 ) a n−1(t)
a n (t)
q n−2 (t) = (t − t 0 ) 2 a n−2(t)
a n (t)
.
q 0 (t) = (t − t 0 ) n a 0(t)
a n (t)
⎫
⎪⎬
⎪⎭
(28.64)
are analytic, then the point is called a regular (or removable) singularity.
If none of the q k (t) are analytic, then the point is called an irregular
singularity. We will need to modify the method at regular singularities,
and it may not work at all at irregular singularities.
For second order equations, we say the a point of t = t 0 is an ordinary
point of
y ′′ + p(t)y ′ + q(t)y = 0 (28.65)
if both p(t) and q(t) are analytic at t = t 0 ; a regular singularity if
(t − t 0 )p(t) and (t − t 0 ) 2 q(t) are analytic at t = t 0 ; and an irregular
singularity if at least one of them is not analytic at t = t 0
Theorem 28.1. [Existence of a power series solution at an ordinary point]
If {p j (t)}, j = 0, 1, ..., n − 1 are analytic functions at t = t 0 then the initial
value problem
}
y (n) + p n−1 (t)y (n−1) + p n−2 (t)y (n−2) + · · · + p 0 (t)y = 0
(28.66)
y(t 0 ) = y 0 , y ′ (t 0 ) = y 1 , ..., y n−1 (t 0 ) = y n