  The classical model of q-damped oscillator is introduced and solved in terms
of Jackson q-exponential function for three different cases, under-damped,
over-damped and the critical one. It is shown that in all three cases solution
is oscillating in time but is unbounded and non-periodic. By q-periodic
function modulation, the self-similar micro-structure of the solution for small
time intervals is derived. In the critical case with degenerate roots, the
second linearly independent solution is obtained as a limiting case of two
infinitesimally close roots. It appears as standard derivative of q-exponential
and is rewritten in terms of the q-logarithmic function. We extend our result
by constructing n linearly independent set of solutions to a generic constant
coefficient q-difference equation degree N with n degenerate roots.
