Exact solutions for certain nonlinear autonomous ordinary differential equations of the second order and families of two-dimensional autonomous systems
Summary: Certain nonlinear autonomous ordinary differential equations of second order are reduced to Abel equations of the first kind ((Ab-1) equations). Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1) equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for most of which only first integrals (in closed or parametric form) have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.
scientific article; zbMATH DE number 5862343