# Power series solution

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• Sep 22nd 2011, 11:28 PM
lindah
Power series solution
http://img263.imageshack.us/img263/5338/70679125.png
Hi guys,

May I confirm if my approach is correct so far as I work towards the solution.

I apologize in advance, it looks a bit long, but it is just the first few steps of the approach (no work on equating coefficients etc yet)

1) This ODE has a regular singular point at x=0 and so Frobenius Method is used where
$y=\sum_{n=0}^{\infty}a_{n}x^{n+s}$?

2) If 1) is correct, when I try to write it out as a series I obtain 3 series:
$x \sum_{n=0}^{\infty} (n+s-1)(n+s)a_{n}x^{n+s-2} - 2 \sum_{n=0}^{\infty} (n+s)a_{n}x^{n+s-1} + 9x^5 \sum_{n=0}^{\infty} a_{n}x^{n+s} &= 0$
Which simplifies to:
$\sum_{n=0}^{\infty} (n+s-1)(n+s)a_{n}x^{n+s-1} - \sum_{n=0}^{\infty} 2(n+s)a_{n}x^{n+s-1} + \sum_{n=0}^{\infty} 9a_{n}x^{n+s+5} &= 0$

May I confirm if this is correct?

3) My next question is that I attempt to make the summation indexes the same - i.e. n=5. However I get a large number of terms that need to be taken out of the first 2 series - so wanted to confirm if my approach is correct in the first place.

Thank you for any feedback given
Linda
• Sep 23rd 2011, 12:22 AM
JJacquelin
Re: Power series solution
2) is correct.
3) Let n=N-4 and rewrite the last sum, starting from N=4. After, the character "N" can be replaced by the character "n".
For information : The EDO is one of Bessel kind. Solutions are :
y = C1 x^(3/2) BesselJ[1/2 , x^3] +C2 x^(3/2) BesselJ[-1/2 , x^3]
• Sep 23rd 2011, 04:21 PM
lindah
Re: Power series solution
Hi JJacquelin,

Thank you for taking the time to look at it and confirm my approach.

I have let n=N-4 and the series solution works out.
May I just confirm if the solution should be:
$y=c_{1}\cos[x^3] + c_{2}\sin[x^3]$?

Thank you!!!