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 $x=e^{st}$, $\huge \begin{align} \ddot x + \frac{\mu}{m}\dot x + \frac{k}{m}x &= 0 \\ \D_{t}^{2}\{e^{st}\} + \frac{\mu}{m} \D_{t}\{e^{st}\} + \frac{k}{m}e^{st} &= 0 \\ s^{2}e^{st}+ \frac{\mu s}{m}e^{st}+\frac{k}{m}e^{st}&= 0 \\ e^{st}\pa{ s^{2} + \frac{\mu}{m}s+\frac{k}{m}} &= 0\\ { s^{2} + \frac{\mu}{m}s+\frac{k}{m}} &= 0 \\ \end{align}$ We can then use the quadratic equation to find the value of $s$ that satisfies this equation $\huge \begin{align} s &= \frac{-b \pm \sqrt{ b^{2} - 4ac }}{2a} \\ \end{align} $