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.