Consider the Riccati equation \begin{equation} y'(x)=P+\left(A\cos(fx-\phi)+Q\right)y(x)^2, \end{equation} for $A,Q\in\mathbb{R}$ and $P,f,\phi\in\mathbb{R}_{>0}$ subject to the initial condition $y(0)=y_0$. Is there an explicit or approximate solution in $\mathbb{C}$? Letting $t=(A\cos(ft-\phi)+Q)y$, assuming a nonzero coefficient, we arrive at \begin{equation} t'=t^2-\left(\frac{Af\sin(fx-\phi)}{A\cos(fx-\phi)+Q}\right)t+(A\cos(fx-\phi)+Q)P. \end{equation} The standard transformation $t=-\Upsilon'/\Upsilon$ will reduce this to a second order ODE with varying coefficients: \begin{equation} \Upsilon''+\frac{Af\sin(fx-\phi)}{A\cos(fx-\phi)+Q}\Upsilon'+P\left(A\cos(fx-\phi)+Q\right)\Upsilon=0. \end{equation} For $A=0$, the solution is straightforward: \begin{equation} y(x)=\sqrt{\frac{P}{Q}}\tan\left(C+x\sqrt{PQ}\right), \end{equation} such that $C\in\mathbb{R}$, which is given by $C=\arctan(y_0\sqrt{Q/P})+n\pi$ for $n\in\mathbb{Z}$ subject to the IC.
There is a rather intractable power series solution for the Riccati equation available here. Perhaps, a simpler expansion can be derived for small $A$ if an explicit solution does not exist in this case.
