Trouble with harmonic oscillator equation

[deex171]

Consider the harmonic oscillator equation (with m=1),
x''+bx'+kx=0
where b≥0 and k>0. Identify the regions in the relevant portion of the bk-plane where the corresponding system has similar phase portraits.

I'm not sure exactly where to start with this one. Any ideas?

 

[jfizzix]

there are certain regions of b and k where the oscillator will undergo overdamped motion, underdamped motion, and critically damped motion. The phase space trajectories of overdamped motion look a lot different than those of underdamped motion. Solving the equation of motion should tell you which values of b and k correspond to which kinds of motion. I expect this is what is being asked.