On the reduction of some second-order nonlinear ODEs in physics and mechanics to first-order nonlinear integrodifferential and Abel’s classes of equations

Second-order ordinary differential equations (ODEs) with strong nonlinear stiffness terms (cubic nonlinearities) governing wave motions, dynamic crack propagations, nonlinear oscillations etc. in physics and nonlinear mechanics are analyzed. Selecting as guide line a second-order nonlinear ODE of the form of the forced Duffing equation and using admissible functional transformations it is possible to reduce it to an equivalent first-order nonlinear integrodifferential equation. The reduced equation is exact. In the limits of small or large values of the parameter characterizing this nonlinear problem, it is shown that further reductions lead to a nonlinear ODE of the Abel classes. Taking into account the known exact analytic solutions of this equivalent equation it is proved that there does not exist an exact analytic solution of this type of equations. However, in cases when convenient functional relations connecting all parameters of the corresponding null equation and the characteristics of the driving force exist, approximate analytic solutions to the problem under consideration are provided.