Abstract
In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.
Highlights
In the research field of periodic solution to Lienard nonlinear differential equations of the form (1.1)x + f (x) = 0 where the overdot stands for the derivative with respect to time, and f (x) is a nonlinear function of x, it is less usual to notice differential equations with exact periodic solutions
In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators
The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations
Summary
In the research field of periodic solution to Lienard nonlinear differential equations of the form (1.1). It is again very less usual to find differential equations with exact periodic solutions in terms of trigonometric functions The existence of periodic solutions of these equations is yet under some debate This becomes an attractive research problem when the second order autonomous truly nonlinear equation has a singularity at the origin and can have no critical point, necessary condition, according to [8], for a planar autonomous systems to have a periodic solution. It was the case of the so-called pseudo-oscillator investigated in [8]. The objective is to calculate the exact and explicit general solutions of the equations (1.3), (1.12) and (1.13)
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