(untitled) (Comprehensive Journal of Science, 2025)
This study examines the possibilities of simple polynomial solutions for specific classes of nonlinear differential equations. It relies on the traditional algebraic approach to solving nonlinear polynomial systems, where projective methods and summations play an important role. The study focuses on first- and second-order nonlinear ordinary differential equations, using the Ansatz method and symbolic computing tools to generate and validate polynomial solutions of order four or less. The main objectives include discovering equations with polynomial solutions, evaluating the effectiveness of analytical methods such as equilibrium analysis and Lie symmetry, and comparing the validity of exact solutions with approximate and numerical alternatives. The research adopts an analytical-theoretical approach, relying on algebraic logic, symbolic software (Mathematica/Maple), and a carefully selected sample of well-known nonlinear models, including the Riccati and Duffing equations. The results demonstrate that, under certain structural and boundary conditions, nonlinear equations can allow for simple polynomial solutions that accurately describe the behavior of the system. The results also demonstrate that the form and degree of nonlinearity of the equation significantly influence the feasibility of developing polynomial solutions. The study emphasizes the importance of symbolic computing in verifying these solutions and suggests further research on higher-order systems with variable coefficients, as well as the integration of polynomial solutions into applicable physical and engineering environments.