  We consider a nonhomogeneous Burgers equation with time variable
coefficients, and obtain an explicit solution of the general initial value
problem in terms of solution to a corresponding linear ODE. Special exact
solutions such as generalized shock and multi-shock solitary waves, triangular
wave, N-wave and rational type solutions are found and discussed. As exactly
solvable models, we study forced Burgers equations with constant damping and an
exponentially decaying diffusion coefficient. Different type of exact solutions
are obtained for the critical, over and under damping cases, and their behavior
is illustrated explicitly. In particular, the existence of inelastic type of
collisions is observed by constructing multi-shock solitary wave solutions, and
for the rational type solutions the motion of the pole singularities is
described.
