Analytical and approximate solutions to autonomous, nonlinear, third-order ordinary differential equations

Analytical solutions to autonomous, nonlinear, third-order nonlinear ordinary differential equations invariant under time and space reversals are first provided and illustrated graphically as functions of the coefficients that multiply the term linearly proportional to the velocity and nonlinear terms. These solutions are obtained by means of transformations and include periodic as well as non-periodic behavior. Then, five approximation methods are employed to determine approximate solutions to a nonlinear jerk equation which has an analytical periodic solution. Three of these approximate methods introduce a linear term proportional to the velocity and a book-keeping parameter and employ a Linstedt–Poincaré technique; one of these techniques provides accurate frequencies of oscillation for all the values of the initial velocity, another one only for large initial velocities, and the last one only for initial velocities close to unity. The fourth and fifth techniques are based on the Galerkin procedure and the well-known two-level Picard’s iterative procedure applied in a global manner, respectively, and provide iterative/sequential approximations to both the solution and the frequency of oscillation.