Example of solving a [[../02 Areas/Math/Differential Equations|Differential Equation]] (specifically [[../02 Areas/Physics/Damped Harmonic Oscillators|Damped Harmonic Oscillators]]) using the [[../02 Areas/Math/Laplace Transform|Laplace Transform]] / [[Complex Exponential]]. $\huge \begin{align} F = m\ddot x &= -\mu \dot x -kx \\ \ddot x + \frac{\mu}{m}\dot x + \frac{k}{m}x &= 0 \\ \end{align}$ Assuming $x=e^{st}$, $\huge \begin{align} \ddot x + \frac{\mu}{m}\dot x + \frac{k}{m}x &= 0 \\ \D_{t}^{2}\{e^{\omega t}\} + \frac{\mu}{m} \D_{t}\{e^{\omega t}\} + \frac{k}{m}e^{\omega t} &= 0 \\ \omega ^{2}e^{\omega t}+ \frac{\mu \omega }{m}e^{\omega t}+\frac{k}{m}e^{\omega t}&= 0 \\ e^{\omega t}\pa{ \omega ^{2} + \frac{\mu}{m}\omega +\frac{k}{m}} &= 0\\ \forall t : e^{\omega t} &\neq 0 \\ { \omega ^{2} + \frac{\mu}{m}\omega +\frac{k}{m}} &= 0 \\ m\omega ^{2} + \mu \omega + k &= 0 \\ \end{align}$ We can then use the quadratic equation to find the value of $s$ that satisfies this equation $\huge \begin{align} \omega &= \frac{-b \pm \sqrt{ b^{2} - 4ac }}{2a} \\ \omega &= \frac{ -\mu \pm \sqrt{ \mu^{2} - 4mk } }{2m} \end{align} $ $\huge \begin{align} x &= \frac{ x_{0} }{2} \pa{e^{\omega_{0} t} + e^{-\omega_{1} t}} \\ x(0) &= \frac{x_{0}}{2} (e^{0t}+e^{-0t}) \\ x(0) &= \frac{x_{0}}{2} (1+1) \\ x(0) &= x_{0} \\ \end{align}$