Abstract
In this paper we study a Lienard equation without restoring force. Although this equation does not satisfy the classical existence theorems, we show, for the first time, that such an equation can exhibit harmonic periodic solutions. As such the usual existence theorems are not entirely adequate and satisfactory to predict the existence of periodic solutions.
Highlights
Where overdot denotes derivative with respect to time, θ(x) and h(x) are arbitrary functions of x, is a generalization of the conservative equation x + h(x) = 0
To extract for the first time, exact and explicit harmonic and isochronous periodic solutions, so that the authors qualified this equation of an unusual Lienard type nonlinear oscillator
Their study shows without ambiguity that the type of equation (1.4) investigated by Chandrasekar et al [7] can exhibit unbounded periodic solutions
Summary
Let f (x) = μ2 − x2, where μ is an arbitrary constant. Using equation (2.7), the solution of equation (3.1) is given by the quadrature, as expected ( [14], [15], [16]), defined in the form dx −a(t + K) =. Where K is an arbitrary constant and C = 0. From the expression of f (x), equation (3.2) takes the form. In this context, by inverting, the exact harmonic and isochronous periodic solution of equation (3.1). Solution (3.5) shows that equation (3.1) and the linear harmonic oscillator x + (−a)2x = 0. As H is a constant, the damped equation (3.1) is a conservative nonlinear system
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