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## Homework Statement

Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0.

## Homework Equations

None.

## The Attempt at a Solution

I'll post my whole work.

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- #1

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Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0.

None.

I'll post my whole work.

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- #5

pasmith

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Recurrences of the form [tex]

a_{n+1} = f(n)a_n[/tex] have the solution [tex]

a_n = a_0\prod_{k=0}^{n-1} f(k)[/tex] where by convention [tex]

\prod_{k=0}^{-1} f(k) = 1.[/tex]

Recurrences of the form [tex]

a_{n+2} = f(n)a_n[/tex] can be turned into the above form by treating even and odd terms separately: First set [itex]n = 2m[/itex] and [itex]b_m = a_{2m}[/itex] to obtain [tex]

b_{m+1} = f(2m)b_m[/tex] and then set [itex]n = 2m+1[/itex] and [itex]c_m = a_{2m+1}[/itex] to obtain [tex]

c_{m+1} = f(2m+1)c_m.[/tex]

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But that's not the answer in the book. How do I get the answer in the book?

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pasmith

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y(x) = \sum_{n=0}^\infty a_nx^n = \sum_{m=0}^\infty b_mx^{2m} + \sum_{m=0}^\infty c_mx^{2m+1}[/tex] with [itex]b_m[/itex] and [itex]c_m[/itex] obtained as I have suggested is not the answer in the book?

What do you get for [itex]b_m[/itex] and [itex]c_m[/itex]?

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