A central oscillator coupled to a bath of harmonic oscillators with a two-dimensional Debye spectrum is set up as a model for the dynamics of strongly coupled linear systems. The bath oscillators are eliminated from the central oscillator's equation of motion, other than for initial conditions. The resulting Langevin equation is solved analytically for two different initial conditions for the bath. In one case, the bath oscillators are started at finite temperature and the coupling is turned on suddenly, and in the other they are adiabatically heated with constant coupling. The problem of equipartitioning of the kinetic energy, the velocity autocorrelation function of the central oscillator, and its spectral distribution are examined for various values of the physical parameters. The analytical results of the sudden case are compared with molecular-dynamics calculations and excellent agreement is found.