Phase space for a damped harmonic oscillator

Question

Given the equation $x''+2 \beta x'+ \omega^2 x=0$ for a damped oscillator, I can get to the equation of motion x(t) and deriving it with respect to time: $x'(t)$.
I am asked to plot the phase plane $\{x,x'\}$, considering the 3 cases: $\beta=0$, $\beta << \omega$, $\omega << \beta$.
I have plotted the phase diagram for an undamped harmonic oscillator ($x''+\omega^2 x=0$) which consisted of ellipses but I can´t find a formula linking $x$ and $x'$.$\\$ I have also tried using the Hamiltonian but arrived nowhere.

Answer 1

$x(t)$ is a function of time, so is $x'(t)$. You should be able to use time to relate the two.