A new approach for solving a complex-valued differential equation

In this paper two analytical approximate solving procedures for a complex-valued differential equation are developed. One of the methods represents the generalization of the Krylov–Bogolubov method for a strong differential equation with complex function. The second method is based on the first integrals of the system. The differential equation is transformed introducing the perturbed first integrals and the polar co-ordinates. The solution is obtained applying the straightforward series expansion. The solution for the special case of without impact initial conditions is considered. The method is applied on the system with strong cubic non-linearity. The small gyroscopic function and damping function are introduced. The analytical approximate solution is compared with numerical exact one and shows a good agreement.