Stability of $y = \cos(2t)$ for the ode $y'' + 4y = 0$ I want to examine the stability of the solution $\phi = \cos(2t)$ of the ode $y'' + 4y = 0$.
I know the general solution of this ode is $y = c_1 \cos(2t) + c_2 \sin(2t)$. So to examine the stability of my solution, I need to see if other solutions stay close to it starting at $t = 0$. It'll be asymptotically stable if other solutions converge to it.
So, when $t = 0$, $\phi(0) = \cos(0) = 1$. and $\phi'(0) = -\sin(0) =0$. But.. how do I pick $y(t)$ so that $y(0)$ starts out close to $1$ and $y'(0)$ is close to $0$? Then what?
Well, $y(t) = 1$ when $t = 0$ and when $t = 0$, $y'(0) = 1$. I'm not sure how to proceed. 

Since all solutions are periodic, this is stable but not asymptotically stable: solutions that start close stay close, but do not converge to your solution.
