Here's a similar, simplified system:
ode = {x''[t] == y'[t]^2 Sin[y[t]] - y''[t] Cos[y[t]] - x'[t],
y''[t] == -1/2 x''[t] Cos[y[t]] - y[t] - Sin[y[t]] - y'[t]};
ics = {x[0] == 2, x'[0] == 1, y[0] == 1, y'[0] == 2};
NDSolve[{ode, ics}, {x, y}, {t, 0, 10}]
