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

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  • #1
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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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  • #2
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This is my work, I have more to post.
 

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  • #3
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This work comes first, the above one comes after this one.
 

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  • #4
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But I don't know how to get to the book's answer.
 

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  • #5
pasmith
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You have the correct recurrence relation [tex]a_{2n} = - \frac{2n+1}{n+1}a_n.[/tex] You just haven't tried to solve it.

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]
 
  • #6
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But that's not the answer in the book. How do I get the answer in the book?
 
  • #7
pasmith
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Are you saying that [tex]
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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