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Ted123

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## Homework Statement

The origin is a regular singular point of the equation [tex]2x^2 y'' + xy' - (x+1)y =0.[/tex] Find 2 independent solutions which are Frobenius series in x.

## The Attempt at a Solution

Substituting [tex]y = \sum_{n=0}^{\infty} a_n x^{n + \sigma}[/tex] eventually gives [tex](2\sigma(\sigma - 1) +\sigma -1 )a_0 x^{\sigma} + \sum_{n=0}^{\infty} \left[ (2(\sigma + n)(\sigma + n+1) + \sigma + n ) a_{n+1} - a_n \right] x^{n+\sigma + 1} = 0.[/tex]

Equating the series to 0 term-by-term gives the indicial equation [tex]2\sigma (\sigma -1) + \sigma -1 = 0 \Rightarrow (2\sigma +1)(\sigma -1) = 0 \Rightarrow \sigma = -\frac{1}{2},\; \sigma = 1.[/tex]

We get the recurrence relation [tex]a_{n+1} = \frac{a_n}{2(\sigma + n)(\sigma + n +1) + \sigma + n}.[/tex]

This is what I'm struggling to solve...

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