Continuous periodic solution of a nonlinear pseudo-oscillator equation in which the restoring force is inversely proportional to the dependent variable

In this paper, we consider a method for the simple exact analytical solution of autonomous nonlinear oscillator equations. While the approach can be used to solve nonlinear oscillator equations with smooth solutions (and we demonstrate this with an application of the approach to an autonomous Duffing equation), our primary interest will be on solving equations with non-smooth yet continuous solutions. To this end, we consider the second-order pseudo-oscillator equation yy″ + 1 = 0 used as a simple model of the path taken by an electron in an electron beam injected into a plasma tube. In recent results of Gadella and Lara, the authors claim the non-existence of periodic solutions to this equation, but actually show that there are no smooth periodic solutions. We show that although there are no smooth solutions to this equation, there does exist a type of continuous periodic solution on the whole problem domain, hence periodic solutions do indeed exist. These periodic solutions can be constructed to have any arbitrary positive period, and the amplitude of these solutions increases as the period is increased. The approach allows one to construct periodic solutions to a variety of nonlinear oscillator equations, even if the solutions are not smooth, and hence could be a useful tool for those interested in physical applications in which nonlinear oscillator models arise