Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden-type equations

The nonlinear generalized modified Emden-type equations (GMEE) are known to be linearizable into simple harmonic oscillator (HO) or damped oscillators (DHO) via some nonlocal transformations. Hereby, we show that the structure of transformation and linearizability HO DHO determines nature/structure dynamical forces involved (hence, determine equation). Yet, a reverse engineering strategy is used so exact solutions emerging GMEE nonlocally transformed find equations. Consequently, whilst solution for remains textbook one, (never reported elsewhere, best our knowledge) turns out manifestly most explicit general offers consistency comprehensive coverage associated under-damping, critical-damping, over-damping cases (i.e. no complex settings coordinates and/or velocities eminent/feasible). Moreover, all system, report illustrative figures each as well corresponding phase-space trajectories they evolve in time.