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

[Esran]

Homework Statement

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

Homework Equations

N/A

The Attempt at a Solution

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

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

Thanks in advance for your help!

 

[Dick]

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.

 

[Esran]

Okay, I see now. How do you compute the distance between them and 2 though, in the complex plane?

 

[Dick]

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