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.