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.
We present in this paper an exceptional Lienard equation consisting of a modified Van der Pol-Helmholtz oscillator equation. The equation, a frequency-dependent damping oscillator, does not satisfy the usual existence theorems but, nevertheless, has an isochronous centre at the origin. We exhibit the exact and explicit general harmonic and isochronous solutions by using the first integral approach. The numerical results match very well analytical solutions.
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