Elementary Quadrature for the Abel and Li\'enard Differential Equations

By appropriate transformation, the problem of solving the Abel equation of the first kind can be transformed into that of solving the quasi-Riccati equation. Using the integrable condition and solution of above quasi-Riccati equation, general solutions of the Abel equation of the first kind in form of elementary quadrature are obtained, which contains numerous. Based on it, the solution of the Li\'enard differential equation can be obtained

Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration