I need to solve the following equation: $$y'(t)+2[\cos y(t)+\Omega(t)]=0,$$ where $$\Omega(t)=-2\eta +\frac{2(\eta^2-1)}{\eta-\cos(4\sqrt{\eta^2-1}t)}$$ with $\eta>1$.
Undoubtedly, the differential equation is so complicated. If it is unsolvable, the problem will be changed, i.e., analyzing the number of solutions for $\sin[y(t)]=1$, where $t\in[t_0,t_0+T]$. Note that $t_0\gg1$ [For the sake of simplicity, we could assume $\sin[y(t_0)]=0$] and $\Omega(t+T)=\Omega(t)$.
First, the differential equation arises from the other differential equations: \begin{align} \dot{m}_y(t)&=2[m_y(t)-\Omega(t)]m_z(t)\\ \dot{m}_z(t)&=2[\Omega(t)-m_y(t)]m_y(t). \end{align} Then, we take advantage of the condition: $m_y^2(t)+m_z^2(t)=1$, and assume the existence of a function $y(t)$ such that \begin{align} m_y(t)&=\cos[y(t)]m_y(0)+\sin[y(t)]m_z(0)\\ m_z(t)&=\cos[y(t)]m_z(0)-\sin[y(t)]m_y(0). \end{align} Consequently, the differential equation about variable $y$ could be obtained when we assume $m_y(0)=-1$ and $m_z(0)=0$.
In practice, $m_y$ and $m_z$ denote as $\langle J_y\rangle/N$ and $\langle J_z\rangle/N$, respectively. Here, $J_y$ and $J_z$ are the collective operators of $N$ quantum spin-1/2 systems, i.e., $J_\alpha=\frac{1}{2}\sum_{i=1}^N\sigma^\alpha_i$, where $\sigma^\alpha$ represents Pauli operator.
