So I just had this problem in my differential equations exam; it started with: Turn this differential equation into a system of first order then find its equilibria and classify them + Lyapunov question $$x''(t)+2\alpha x'(t)+\alpha^2\sin (x(t))=0$$
The problem was I could't get rid of the $\sin(x)$ so I said for $\sin(x)\approx x$ do $$z'=\begin{pmatrix}z_1\\z_2 \end{pmatrix}=\begin{pmatrix}x''\\ x'\end{pmatrix}=\begin{pmatrix}-2\alpha&-\alpha^2\\ 2\alpha&0\end{pmatrix}\begin{pmatrix}x'\\ x\end{pmatrix}$$ but this seems wrong too. How do I do this?#
So Yeah turns out I read system as linear system so that's that
Set $y(t):=x'(t)$ and
$$\begin{cases}x'(t)=y(t),\\y'(t)=-2\alpha y(t)-\alpha^2\sin(x(t))\end{cases}$$ is your system.