Stability of a simple ODE

I have the following ODE:

Where A,w are constants strictly greater than 1.

I am asked to find the solution of the ODE and derive the criterion of stability of the point X = 0 in terms of the parameters A and w.

I can do the solving, I just don't understand what the whole stability thing is asking. I thought stability occured at fixed points, ie: where X'=0 not X=0. What am I supposed to do?

Here's what I've done so far:

To find the solution I let and then solve the equation

This yields an auxiliary equation of with solutions

Case One:

b>0 then we have solution

Case Two:

b=0 then

Case Three:

b<0 then

(where alpha and beta are constants in all cases)

How do I actually derive this supposed criterion of stability for X=0? Any help would be greatly appreciated, I need to touch up on my differential equations, it's been a reasonably long time since I last studied them.