find the good series solution to the ODE

This is what I did:

So, is such that . Therefore, is an ordinary point for all real value of .

We want a solution like

If we substitute in (1), we have

because doesn't change anything.

Let then the relation is given by :

or

Let's find the first terms:

etc.

But I can't find a better solution than

Can I improve it?

Best Regards,

Fractalus

Re: find the good series solution to the ODE

Your solution seems to be 'all right'!...A simple approach however for finding a solution of the ODE ...

(1)

... is to search for a search of the coefficients of the Taylor expansion...

(2)

The initial conditions give us and . The other derivatives are derived from (1)...

(3)

(4)

(5)

(6)

(7)

Now You are able to write the Taylor expasion till to the order 6...

(8)

Of course You can proceed to higher orders...

Kind regards

Re: find the good series solution to the ODE

Quote:

Originally Posted by

**chisigma** Your solution seems to be 'all right'!...A simple approach however for finding a solution of the ODE ...

(1)

... is to search for a search of the coefficients of the Taylor expansion...

(2)

The initial conditions give us

and

. The other derivatives are derived from (1)...

(3)

(4)

(5)

(6)

(7)

Now You are able to write the Taylor expasion till to the order 6...

(8)

Of course You can proceed to higher orders...

Kind regards

Thanks chisigma,

I computed a solution with Mathematica and he gave me that:

Taken from:

http://www.wolframalpha.com/input/?i=y%27%27+%2B+t^2y%27+%2B+2ty+%3D+0

I think that I should have been able to find but I can't figure out how with what I have. The Taylor Series Expansion of that is probably something like