(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In this study, a fourth-order nonlinear ordinary differential equation is considered. The specificity of nonlinearity lies in the presence of moving singular points, which hinders the application of classical theory that only works in the linear case. Two research problems are addressed in this work: the theorem of existence and uniqueness of the solution, and the precise criteria for the existence of a moving singular point. These problems are solved in both the real and complex domains. The specificity of transitioning to the complex plane is demonstrated using phase spaces. The obtained results are validated through numerical experiments, confirming the reliability of the results.