$\huge \begin{align} m \ddot x &= -b\dot x -kx \\ m \ddot x + b \dot x+kx &= 0 \\ \end{align}$ $ \huge \let x = Ae^{st} $ $\huge \begin{align} m\D^{2}\set{Ae^{st}} + b\D\set{Ae^{st}}+kAe^{st} &= 0\\ Ams^{2}e^{st} + Abse^{st}+Ake^{st} &= 0 \\ Ae^{st}\pa{ms^{2}+bs+k} &= 0 \\ \nexists s \in \C : e^{st} &= 0 \\ ms^{2}+bs+k &= 0\\ \end{align}$ $\huge \begin{align} s&= \frac{-b \pm \sqrt{ b^{2} - 4mk }}{2m} \\ s&= -\frac{b}{2m} \pm \frac{\sqrt{ b^{2}-4mk }}{ \sqrt{ (2m)^{2} }} \\ s&= -\frac{b}{2m} \pm \frac{\sqrt{ b^{2}-4mk }}{ \sqrt{ 4m^{2} }} \\ s &= -\frac{b}{2m} \pm \sqrt{ \frac{b^{2}}{4m^{2}} - \frac{4mk}{4m^{2}} } \\ s &= -\frac{b}{2m} \pm \sqrt{ \frac{b^{2}}{4m^{2}} - \frac{k}{m} } \\ \end{align} $ $\huge \begin{align} \omega &= \sqrt{ \frac{b^{2}}{4m^{2}} - \frac{k}{m} } \\ x&= \frac{x_{0}}{2} e^{t\left( -\frac{b}{2m} + \omega \right)} + \frac{A_{0}}{2} e^{t\pa{ - \frac{b}{2m} - \omega }} \\ x&= \frac{x_{0}}{2} e^{- \frac{-bt}{2m}}\pa{e^{\omega t}+e^{-\omega t}}\\ x&= \frac{x_{0}}{2} e^{- \frac{-bt}{2m}}\pa{ \cos (\omega t) +i\sin(\omega t) +\cos(\omega t) + i\sin(-\omega t)) }\\ x&= \frac{x_{0}}{2} e^{- \frac{-bt}{2m}}\pa{ 2\cos(\omega t)+i\sin(\omega t) -i\sin(\omega t) }\\ x&= \frac{x_{0}}{2} e^{- \frac{-bt}{2m}}\pa{ 2\cos(\omega t) }\\ x&= x_{0}e^{- \frac{bt}{2m}}\cos(\omega t) \end{align} $ Assuming this is an underdamped system, $\huge \frac{k}{m} > \frac{b^{2}}{4m^{2}} $ $\huge \begin{align} \let \omega &= \sqrt{ \frac{b^{2}}{4m^{2}} - \frac{k}{m} } \\ x &= e^{-\frac{bt}{2m} }\pa{ \cos \omega t + i\sin \omega t } \end{align} $