The first and second derivatives are: \[\begin{align} \frac{dz}{dt} &= -i\omega\, e^{-i\omega t} \\ \frac{d^2z}{dt^2} &= -\omega^2\, e^{-i\omega t}\end{align}\] Substituting these into the differentia...The first and second derivatives are: \[\begin{align} \frac{dz}{dt} &= -i\omega\, e^{-i\omega t} \\ \frac{d^2z}{dt^2} &= -\omega^2\, e^{-i\omega t}\end{align}\] Substituting these into the differential equation gives: \[\left[-\omega^2 - 2i\gamma \omega + \omega_0^2 \right] e^{-i\omega t} = 0.\] This equation holds for all \(t\) if and only if the complex second-order polynomial on the left-hand side is zero: \[-\omega^2 - 2i\gamma \omega + \omega_0^2 = 0.\] The solutions for \(\omega\) can be …
