Math Is Fun Forum

lindah

Member

Registered: 2010-01-25

Posts: 121

Power Series solution

Hi guys,

Apologies for the bombardment of DE's

I just wanted to see if my setup is correct before continuing, so I hope you can give me feedback :

1) This ODE has a regular singular point at x=0, and so Frobenius Method can be applied with

?

2) If the above is correct I have the ODE in series form simplified as:

3) Rewriting them with the common power

I obtain

4) I ask because I am concerned about the next step, as this would mean I would need to take out the first 6 terms of the first 2 series, meaning I would get many indicial roots for (s)

Thank you in advance for any feedback
Linda

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Power Series solution

Hi Linda;

First, take a look here:

http://math.fullerton.edu/mathews/n2003 … esMod.html

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Power Series solution

Hi bobbym,

Thank you for the link! It has some nice Mathematica code

Most of the Frobenius examples (in the link, my text and course notes) usually only deal with indicial equations
where its for e.g.

and so its just quadratic roots

I've been running into incidents where the indicial  would be something along the lines:

Implying:

I am thinking, find the roots of the first equation, meaning

?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Power Series solution

Hi Linda;

Maybe, you might get some like that but this indicial equation is only a quadratic.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Power Series solution

Hi bobbym;

Sorry, I meant that we would deal with more than one quadratic

Extracting the terms for this was:

So

and

all equal 0.

Thanks for walking me through this!
Done and dusted,

Linda

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Power Series solution

Hi;

Thanks, but you did all the work.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

zetafunc.

Guest

Re: Power Series solution

If you are interested, another problem that can be solved using infinite series is the solution to Airy's equation (y'' = xy). Can't remember what the initial conditions are (can probably be found from a google search), but there is a nice solution using Taylor series.

zetafunc.

Guest

Re: Power Series solution

(this equation has many applications such as modelling the refraction of light)

lindah

Member

Registered: 2010-01-25

Posts: 121

Re: Power Series solution

Hi zetafunc.

Thank you for the suggestion!!! I am currently working with Airy and Biry functions, as well as Bessel functions.
In setting the assignment, my lecturer has decided to throw curve balls to put us in our place!!!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Power Series solution

Well good for him/her. That will teach those snotty students a good lesson. My hat would go off to him/her if I had a hat. What is wrong with a little berating of your students? It is better than a poke in the eye from a sharp stick isn't it?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.