DSolve[{t^2*(1 - Sin[t])*Derivative[1][x][t] == t*(1 - 2*Sin[t])*x[t] + t^2*y[t], t^2*(1 - Sin[t])*Derivative[1][y][t] == (t*Cos[t] - Sin[t])*x[t] + t*(1 - t*Cos[t])*y[t]},{x[t], y[t]},t]