# Homework Help: Stability equibrilium solution

1. Dec 21, 2008

### dirk_mec1

1. The problem statement, all variables and given/known data
Determine the stability property of the following equibrilium solution:

(0,0) of

$$\ddot{x}+ \alpha x +x=0$$ with $$\alpha \in \mathbb{R}$$

2. Relevant equations
- A solution is Lyapunov stable if for each $\epsilon$ there is a $\delta$ such that: $$||x(0)|| \leq \delta$$ yields $$||x(t)|| \leq \epsilon$$

- A solution is unstable if it isn't stable.

- A solution is asymptotically stable if there is a delta such that: $$||x(0)|| \leq \delta$$ yields $$\lim_{t \rightarrow \infty} ||x(t)|| = 0$$

Note: x can be a vector.

3. The attempt at a solution

I don't know how to use these defintions. Can I just use the 2-norm? Can someone explain me how this works?