n-dimensional PDM-damped harmonic oscillators: linearizability, and exact solvability

We consider position-dependent mass (PDM) Lagrangians/Hamiltonians in their standard textbook form, where the long-standing \emph{gain-loss balance} between kinetic and potential energies is kept intact to allow conservation of total energy (i.e., $L=T-V$, $H=T+V$, $dH/dt=dE/dt=0$). Under such settings, we discuss report on $n$-dimensional PDM damped harmonic oscillators (DHO). use some point canonical transformation facilitate linearizability $n$-PDM dynamical equations into $n$-linear DHOs' for constant setting. Consequently, well know exact solutions linear DHOs are mapped, with ease, onto DHOs. A set one-dimensional a PDM-DHO illustrative examples reported along phase-space trajectories.