Lecture Notes in Differential Equations - Bruce E. Shapiro

273
Summary of Power series method.
To solve the linear differential equation
L n y = a n (t)y (n) + · · · + a 0 (t)y = f(t) (28.136)
or the corresponding initial value problem with
as a power series about the point t = t 0
1. Let y =
∞∑
c k (t − t 0 ) k
k=0
y(t 0 ) = y 0 , ..., y (n−1) (t 0 ) = y n (28.137)
2. If initial conditions are given, use Taylor’s theorem to assign the first
n values of c k as
c k = y (k) (t 0 )/k! = y k /k!, k = 0, 1, ..., n − 1 (28.138)
3. Calculate the first n derivatives of y.
4. Substitute the expressions for y, y ′ , ..., y (n) into (28.136).
5. Expand all of the a n (t)that are not polynomials in Taylor series about
t = t 0 and substitute these expansions into the expression obtained
in step 4.
6. Multiply out any products of power series into a single power series.
7. By an appropriate renumbering of indices, combine all terms in the
equation into an equation of the form ∑ k u k(t − t 0 ) k = 0 where each
u k is a function of some set of the c k .
8. Use linear independence to set the u k = 0 and find a relationship
between the c k .
9. The radius of convergence of power series is min {ti}(|t 0 − t i )) where
{t i } is the set of all singularities of a(t).