Numerically this solution would be straightforward. Your equations appear to come from a model of a physical system --probably a series of masses, springs and dampers -- am I correct?
If this is the case, I would recommend looking at Wolfram's SystemModeler. It is a much easier way to work with physical systems and it integrates well with Mathematica. You could arrange the blocks in a physical way, change constants from SystemModeler or Mathematica, get an animation of your system for visualization, and even run sensitivity analysis on your constants (see which ones have the biggest effect on some output).