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Lower bound for radius of convergence of series solutions about a given point

  1. Dec 9, 2009 #1
    1. The problem statement, all variables and given/known data

    Determine a lower bound for the radius of convergence of series solutions about a) [tex]x_{0}=0[/tex] and b) [tex]x_{0}=2[/tex] for [tex]\left(1+x^{3}\right)y''+4xy'+y=0[/tex].

    2. Relevant equations

    N/A

    3. The attempt at a solution

    The zero of [tex]P\left(x\right)=\left(1+x^{2}\right)[/tex] is [tex]-1[/tex]. The distance between [tex]-1[/tex] and [tex]0[/tex] is [tex]1[/tex], so a) [tex]1[/tex]. The distance between [tex]-1[/tex] and [tex]2[/tex] is [tex]3[/tex], so b) [tex]3[/tex].

    The only problem is that for b), the back of the book gives [tex]\sqrt{3}[/tex]. What am I doing wrong?

    Thanks in advance for your help!
     
  2. jcsd
  3. Dec 9, 2009 #2

    Dick

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    You mean P(x)=(1+x^3). In fact, that has three roots. Two of them are complex, but you have to consider them also. A couple of them are a distance sqrt(3) from x=2.
     
  4. Dec 9, 2009 #3
    Okay, I see now. How do you compute the distance between them and 2 though, in the complex plane?
     
  5. Dec 9, 2009 #4

    Dick

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    One of the roots is cos(pi/3)+i*sin(pi/3), right? Compute the distance from that point to 2+0i.
     
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