Note to Moderator: This Thread should be added to the SystemModeler Group.

I put together a quick example to get you started. I used the translational blocks -- they do not animate. If you want an animating example I can quickly put one together for you. The force is turned on at 2 seconds. By modifying this model you should be able to do whatever you want. Note-- I made the second mass always two times the first mass (but you can change this to a number or an equation). The attached file is the full model for SystemModeler. Here is a picture of what I did:

The simulation results:

Attachments:

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).

I would recommend solving it numerically with NDsolve. Is there a reason you need an analytical solution? I doubt that you will ever get one given the complexity of the equations. Numerically your equations will be easy to solve.