Math Help - Easy Question

1. Easy Question

$(x^2+2)y'' + 2xy' = 0$

Verify $x=0$ is an ordinary point. Then find a lower bound for the radius of convergence of power series solutions about $x=0$

2. Originally Posted by Creebe
$(x^2+2)y'' + 2xy' = 0$

Verify $x=0$ is an ordinary point. Then find a lower bound for the radius of convergence of power series solutions about $x=0$
If you put it in standard form you get

$y'' +\frac{2x}{x^2+2}y'=0$

Since the limit as $\lim_{x \to 0}\frac{2x}{x^2+2}=0$ is finite it is an ordinary point.

The power series will have radius of convergence of at least the distance to the nearest singularity in the complex plane.

$x^2+2=0 \iff x \pm i\sqrt{2}$

So the radius of convergence of at least $R=\sqrt{2}$