Harmonic and Nonperiodic Solutions of Velocity-Dependent Conservative Equations

In this paper, we study a velocity-dependent quadratic Helmholtz equation presumed to be a conservative oscillator with exact harmonic solutions when the Hamiltonian of the system is zero with specific initial conditions. In contrast, we exhibit general harmonic and isochronous solutions for a nonzero value of the Hamiltonian of the system using the first integral method. We also prove the existence of nonperiodic solutions that have not been shown in earlier works. Consequently, the so-called velocity-dependent conservative nonlinear oscillator investigated is simply and purely a pseudo-oscillator.