Graphical Portraits of the Solutions of Binary First Order Nonlinear Ordinary Differential Equation Near Their Singular Point

This paper deals with the appropriate normal homeomorphic forms of a nonlinear binary first order ordinary differential equation (ODE) with smooth coefficients near the critical (singular) point (0, 0) . The normal forms (A. Davydov, then T. Fukui) depend on a real parameter \( \lambda \) and can be semicubic parabola, folded saddle point, folded node and folded focus. The corresponding graphical portraits in the plane are proposed in Theorem 2 and are illustrated geometrically by several figures.