eqn1 = m1 x1''[t] + k1 (x1[t] - x2[t]) + c1 (x1'[t] - x2'[t]) == 0;
eqn2 = m2 x2''[t] + k1 (x2[t] - x1[t]) + c1 (x2'[t] - x1'[t]) + k2 (x2[t]) + c2 (x2'[t])
==c2*(1.09013 Cos[Pi*13.88*t]) + k2*(0.025 Sin[Pi*13.88*t]);

ivals = {x1[0] == 0, x1'[0] == 0, x2[0] == 0, x2'[0] == 0};
soln = NDSolve[{{eqn1, eqn2}, ivals}, {x1[t], x2[t]}, {t, 0, 10}];

eqn1 = x1''[t] == - (x1[t] - x2[t]) - (x1'[t] - x2'[t]);
eqn2 = x2''[t] == -k1 (x2[t] - x1[t]) - (x2'[t] - x1'[t]) + - (x2[t]);

ivals = {x1[0] == 1, x1'[0] == 0, x2[0] == 1, x2'[0] == 0};
 sol = DSolve[Flatten@{{eqn1, eqn2}, ivals} // Evaluate, {x1[t], x2[t]}, t]

eqn1 = m1 x1''[t] == - k1 (x1[t] - x2[t]) - c1  (x1'[t] - x2'[t]);
eqn2 = m2 x2''[t] == -k1 (x2[t] - x1[t]) -  c2 (x2'[t] - x1'[t]) + - (x2[t]);