Harmonic and non-periodic solutions of velocity-dependent conservative equations

Abstract We study in this paper a quadratic damping Helmholtz equation presumed to be velocity-dependent conservative nonlinear oscillator. We show that under the usual conditions of existence of particular and exact harmonic solutions, the equation can also exhibit exact and general non-periodic solutions. We show finally the existence of exact and explicit general harmonic and isochronous solutions without requiring that the system Hamiltonian must be identically zero.