(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine a series solution to the following ODE about x_{0}= 0:

[tex]xy'' + y' + xy = 0[/tex]

3. The attempt at a solution

I'll try to keep this concise.

I first divided through by x and made the usual guesses for the form of the series. Subbing those in gave:

[tex]\sum_{2}^{\infty}n(n-1)a_{n}x^{n-2}+ \sum_{1}^{\infty}na_{n}x^{n-2} + \sum_{0}^{\infty}a_{n}x^{n}[/tex]

Then I shifted the first two series up 1 and the third one down 1 and multiplied through by x:

[tex]\sum_{1}^{\infty}n(n+1)a_{n+1}x^{n}+ \sum_{0}^{\infty}(n+1)a_{n+1}x^{n} + \sum_{1}^{\infty}a_{n-1}x^{n}[/tex]

Then to get the second series to start at 1, I moved the lower index up 1 and added the n = 0 term to make up for it:

[tex]\sum_{1}^{\infty}n(n+1)a_{n+1}x^{n}+ \sum_{1}^{\infty}(n+1)a_{n+1}x^{n} + \sum_{1}^{\infty}a_{n-1}x^{n}+a_{1}[/tex]

Now you can combine the series:

[tex]\sum_{1}^{\infty}[n(n+1)a_{n+1}+(n+1)a_{n+1}+a_{n-1}]x^{n} + a_{1}[/tex]

But then setting coefficients equal to zero gives a_{1}= 0, but don't you need to determine recurrence relations for a_{0}and a_{1}?

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# Series Solution to ODE

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