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No external force:  homogeneous equation

{$mu''(t) + \gamma u'(t) + ku(t) = 0, ~~m, \gamma, k \geq 0$}

$mr^2 + \gamma r + k = 0$
implies
$r_1, r_2 = {-\gamma \pm \sqrt{\gamma^2 - 4km} \over 2m}$

Case 1: ${\gamma^2 - 4km} < 0$

$r_1, r_2 = {-\gamma \pm \sqrt{\gamma^2 - 4km} \over 2m} = 
{-\gamma \pm \sqrt{(-1)(4km - \gamma^2)} \over 2m} = 
{-\gamma \pm i \sqrt{4km - \gamma^2} \over 2m}$


$u(t) = e^{-{\gamma t \over 2m}}(A cos \mu t + B sin \mu t)   $
$= e^{-{\gamma t \over 2m}}R cos( \mu t - \delta)   $

where $\mu =  {\sqrt{4km - \gamma^2} \over 2m}$

and $A = Rcos(\delta)$,  $B = Rsin(\delta)$ 
%%\hfil \break
 ($R = \sqrt{A^2 + B^2}$, $tan{\delta} = {B \over A})$

Note $A, B$ and hence $R, \delta$ depend on initial conditions.  

$\mu$ does not depend on initial conditions \hfil \break
i.e., $\mu$ = quasi frequency and ${2\pi \over \mu}$ = quasi period do not depend on initial conditions. 
\vskip 9pt
\hrule

$\gamma = 0$:  No damping

$u(t) = A cos \mu t + B sin \mu t = R cos( \mu t - \delta)$


$\mu =  {\sqrt{4km} \over 2m} = \sqrt{k \over m} = \omega_0 $




Example:  $y'' + 9y = 0$ 



With damping:  $\gamma \not= 0$, 

Small damping:  $\gamma < 2\sqrt{km}$ ~~(i.e., ${\gamma^2 - 4km} < 0$).


$y'' + y' + 9y = 0$

$y'' + 3y' + 9y = 0$


$u(t) = e^{-{\gamma t \over 2m}}(A cos \mu t + B sin \mu t)   $
$= e^{-{\gamma t \over 2m}}R cos( \mu t - \delta)   $

where $\mu =  {\sqrt{4km - \gamma^2} \over 2m}$

and $A = Rcos(\delta)$,  $B = Rsin(\delta)$ 
%%\hfil \break
 ($R = \sqrt{A^2 + B^2}$, $tan{\delta} = {B \over A})$

Note $A, B$ and hence $R, \delta$ depend on initial conditions.  $\mu$ does not depend on initial conditions. 


Case 2:  Critical Damping:  $\gamma = 2\sqrt{km}$


 $\gamma^2 - 4km = 0$:  $u(t) = (A + Bt)e^{r_1t}$

$y'' + 6y' + 9y = 0$

Case 3:  Over-damped: $\gamma > 2\sqrt{km}$


$\gamma^2 - 4km > 0$:  $u(t) = Ae^{r_1t} + Be^{r_2t}$

$y'' + 7y' + 9y = 0$

$y'' + 10y' + 9y = 0$


\end