Starting from RWA Hamiltonian,
$$ \hat{H}_{\rm RWA}=\frac{\hbar}{2} \begin{bmatrix} 0 && \Omega \\ \Omega^* && -2\Delta \end{bmatrix} $$
TDSE is written as
$$ i\hbar \partial_t\begin{bmatrix} c_g \\ c_e\end{bmatrix}=\frac{\hbar}{2} \begin{bmatrix} 0 && \Omega \\ \Omega^* && -2\Delta \end{bmatrix}\begin{bmatrix} c_g \\ c_e\end{bmatrix} $$
Let’s assume two parameters $(\Omega, \Delta)$ are constant.
If the detuning is zero $\Delta=0$,
$$ \dot{c}_g = -i\frac{\Omega}{2} c_e \\ \dot{c}_e = -i\frac{\Omega^*}{2}c_g $$
The 2nd order differential equation is given from the above relations
$$ \ddot{c}_g +\frac{|\Omega|^2}{4}c_g=0 $$
By assuming the initial condition, $c_g(t=0)=1$ and $c_e(t=0)=0$, the solution is driven as
$$ c_g(t) = \cos\left(\frac{|\Omega|}{2}t\right) \\
c_e(t) = -i\sin\left(\frac{|\Omega|}{2} t\right) $$
The probability of each state becomes,
$$ P_g(t) = |c_g(t)|^2 = \frac{1}{2}+\frac{1}{2}\cos(|\Omega| t) \\
P_e(t) = |c_e(t)|^2 = \frac{1}{2}-\frac{1}{2}\cos(|\Omega| t) $$
The oscillation frequency of each probability is $\Omega$, that’s why $\Omega$ is called Rabi frequency. This light-induced probability oscillation is Rabi oscillation.
In the generalized case,
$$ \dot{c}_g = -i\frac{\Omega}{2} c_e \\ \dot{c}_e = -i\frac{\Omega^*}{2}c_g+i\Delta c_e $$
The 2nd order differential equation is given from the above relations
$$ \ddot{c}_g -i\Delta\dot{c}_g+\frac{|\Omega|^2}{4}c_g=0 $$