Hi guys I am trying to prove that the solution for $x''+x=0$ has a solution that is stable, but not asymptotically stable. The hardest part of the proof is that we never actually defined stable for anything other than first order equation.
Definition: The solution $x=0$ is stable if for all $\epsilon >0$ there exists $\delta$ such that if $x(t)$ is a solution of $x'=Ax$ where A is a constant matrix and $x(0)=x_0$ and $||x||<\delta$ implies that $||x(t)||< \epsilon$ for all $t$.
My thinking is that for second order differential equation we need to let that $x(0)=\delta$ and $x'(0)= \delta_1$ then just look at the abs of the solution $|x(t)|=| \delta cos(t)+\delta_1 sin(t)| \leq \delta + \delta_1$. Am I correct? Also I think this is not asymptotically because as we take $t \rightarrow \infty$ it does not go to zero. Also can someone please actually help me see the definition for higher order differential equations, so far I was unable to see it.