A $1$ kg body is attached on a spring. The initial displacement is $x_0=0.5$m then we drop it. The equation which describes movement of mass is:
$\ddot x+3\dot x + 2x=0$.
I have managed to solve the differential equation and got:
$x=-0.5e^{-2t}+e^{-t}$
Question is: will mass oscillate and what will be it's maximum displacement from equilibrium point?
What I've done:
I calculated motion amplitude ($A^2$=$c_1^2+c_2^2$ $c_1$=-0.5 and $c_2=1$) and my answer is that it will oscillate since it's not critical damping.
I don't think that it's allowed what I've done (I'm not sure if equation for amplitude holds for damped vibration and if amplitude really is maximum displacement). I am also not sure how to find out whether system will oscillate.
I don't remember all the physics words for this underdamped, overdamped et.c. It was a long time ago I did it. But with Laplace/Fourier transforms you can derive that solving the second order polynomial equation
$$ar^2+br+c=0 \hspace{1cm}(\text{ solve for r })$$
will help you if you want to solve the differential equation
$$ay''+by'+c=0$$
The solution to a second order polynomial you can get by formula or with completing the square.
If you know of complex numbers, then the magnitude of the solutions $r_1,r_2$ determines the real exponent and the argument of the solution determines the frequency of oscillation.