DSolve[{k xk + k xa[t] + m xa''[t] ==
   F0 Cos[t [Omega]] + k xb[t],
  2 k xb[t] + m xb''[t] == k (xa[t] + xc[t]),
  m xc'')[t] == k (xk + xb[t] - xc[t]), xa'[0] == 0,
  xb'[0] == 0, xc'[0] == 0, xa''[0] ==  0, xb''[0] ==  0,
  xc''[0] ==  0, xa[0] == 0, xb[0] ==  xk, xc[0] == 2 xk}, {xa[t],
  xb[t], xc[t]}, t].