# Phase space for a damped harmonic oscillator [closed]

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.

• Concerning Lagrangian & Hamiltonian formulations, see e.g. this Phys.SE post. – Qmechanic Dec 16 '20 at 17:46
• What about the answer looks problematic to you? – Cosmas Zachos Dec 18 '20 at 15:05

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