Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 28
Series Solutions
In many cases all we can say about the solution of
a(t)y ′′ + b(t)y ′ + c(t)y = f(t)
y(t 0 ) = y 0
y ′ (t 0 ) = y 1
⎫
⎪⎬
⎪ ⎭
(28.1)
is a statement about whether or not a solution exists. So far, however, we
do not have any generally applicable technique to actually find the solution.
If a solution does exist, we know it must be twice differentiable (n times
differentiable for an nth order equation).
If the solution is not just twice, but infinitely, differentiable, we call it an
analytic function. According to Taylor’s theorem, any function that is
analytic at a point t = t 0 can be expanded in a power series
where
y(t) =
∞∑
a k (t − t 0 ) k (28.2)
k=0
a k = y(k) (t 0 )
(28.3)
k!
Because of Taylor’s theorem, the term analytic is sometimes used to mean
that a function can be expanded in a power series at a point (analytic ⇐⇒
infinitely differentiable ⇐⇒ power series exists).
The method of series for solving differential equations solutions looks
for analytic solutions to (28.1) by substituting equation (28.2) into the
differential equation and solving for the coefficients a 0 , a 1 , ...
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