Singularities of quasi-linear differential equations

We study solutions of quasi-linear ordinary differential equations of the second order at their singular points, where the coefficient of the second-order derivative vanishes. Either solutions entering a singular point with definite tangential direction (proper solutions) or those without definite tangential direction (oscillating solutions) are considered. It is shown that oscillating solutions generically do not exist, and proper solutions enter a singular point in strictly definite tangential directions. A local representation for proper solutions in a form similar to Newton–Puiseux series is obtained.