I have an exam in about 2 hours and I just want to know the steps involved in converting $$\ddot{x}+2k\dot{x}+\omega^2_0x=0$$ into two first order differential equations.
I need to be able to draw its trajectories in a phase plane.
I'm thinking you re-write $$\ddot{x}=-2k\dot{x}-\omega^2_0x$$
Let $y=\dot{x}$ we have
$$\dot{y}=-2k{y}-\omega^2_0 x$$ $$\dot{x}= y $$
How do I write that as a matrix and compute the eigenvalues? Do I have to write this
$$\begin{pmatrix} 0 && 1 \\ -\omega^2_0 && -2k \\ \end{pmatrix}$$
or can I write this instead
$$\begin{pmatrix} 0 && 1 \\ -1 && -2 \\ \end{pmatrix}$$
I need to be able to compute the eigenvalues so I can draw a phase portrait. Any advice is much appreciated