Method of Power Series¶
Lots of ODE cannot be sovled using elementary integration method, so we have to give up solution of finite form and try to find solution of infinite form, like series.
To have a more global understanding, we focus on
where
where
To simplify the notation, we denote
denote
Since
where
Excellent Series¶
To prove that the solution is analytic, we have to use a method of proof, that is, using Excellent Series.
Definition of Excellent Series
Assume there are two power series
and
If
Then we call series
then we call its summing function
引理1: 解析函数有定义域收缩的优函数 | Lemma 1: A analytic function has an excellent function within smaller region
If
then
is an Excellent function of
Use Abel Second Theorem. Note that we can not guarrantee the convergence on boundaries, so we choose open intervals.
We can represent
in region
is convergent, so each item of the above series can be bounded by a number
Now, the following thing is a little tricky. Define
Consider a power of series
which is convergent because it can add up to:
with its range of definition
With the definition of Excellent function, we have to use it to formulate an excellent series of the original series. This is the following theorem.
引理2: 用上述优函数建立的微分方程有解析解 | Lemma2: ODE combined with The above Excellent Function has a solution that can be represented by Power Series
Cauchy problem
has a analytic solution
use elementary integration method.
We let
where
(Here readers can see that
The above ODE is a variable separation equation. So change the form and integrate on
get
That is,
We want to use this form to get a power series. See that
choose
So solution of the above ODE
can be represented by power series of
证明 | Proof¶
Cauchy 定理 | Cauchy Theorem
Assume
-
Represent solution with power series. Show that it is unique.
-
Use an excellent series to prove the above power series convergent.
- Represent
with power series
And represent solution with power series
substitute
and get
Denote
and get
Generally, we have
where
We leave the proof of this part as an additional work in Appendix at the end of the doc.
- Prove the above series converges.
Here we formulate another ODE and use Excellent function to bound the above power series.
Since
If we represent both of them in terms of power series
then we have a relation
Now consider an ODE
by Lemma 2, the above ODE has an analytic solution
Similarly, we have
So power series
附录 | Appendix: Relation between and ¶
where
Use induction.