I need help in solving the following differential equation for $z$, a complex number, and $a,A,B,C$ all real: $$\frac{dz}{dt}=4ia(A\bar{z}+B\bar{z}^2+C\bar{z}^3)$$
I have tried the ansatz $z=ce^{mt}$ with $c,m$ complex but that gives $m=0$. Also trying $z=r(t)e^{i\theta}$ gave me equations I couldn't solve.
For $B=C=0$, the equation is: $$\frac{dz}{dt}=4iaA\bar{z}$$ $$\implies mc=4iaA\bar{c}e^{(m-\bar{m})t}$$ This gives $\bar{m}=m$ and $m=4iaA\bar{c}/c$ If we take $c=re^{i\theta}$, $m=4iaAe^{-2i\theta}$. Since $m$ has to be real, $cos(2\theta)=0$ and thus the system can be solved exactly.