I was reading QUANTUM OPTICS for BEGINNERS by Z FICEK, in his discussion of the Jaynes-Cummings model he use Laplace transform method to solve two coupled differential equations (8.9) as shown in the picture. I tried the method but i never arrive to the same result.
Thus, we have obtained two coupled differential equations that can be written in the form: $$\begin{cases} \dot{C}_{1n}(t)=-in\omega_0C_{1n}(t)+\frac12g\sqrt{n}C_{2n}(t) \\ \dot{C}_{2n}(t)=-in\omega_0C_{2n}(t)-\frac12g\sqrt{n}C_{1n}(t) \\ \end{cases} \tag{8.9}$$ The set of equations $(8.9)$ can be solved for arbitrary initial conditions using, for example, the Laplace transform method. The solution for the amplitude $C_{2n}(t)$ is given by $$C_{2n}(t) =\frac12e^{-in\omega_0t}\{[iC_{2n}(0)-C_{1n}(0)]e^{\frac12 i\Omega t} + [iC_{2n}(0)+C_{1n}(0)]e^{-\frac12 i\Omega t}\} \tag{(8.10)} $$, where $\Omega = g\sqrt{n}$ is the Rabi frequency.