DSolve[{44*x[t] + 49*y[t] + 4*Derivative[1][x][t] + 9*Derivative[1][y][t] == t, 34*x[t] + 38*y[t] + 3*Derivative[1][x][t] + 7*Derivative[1][y][t] == E^t},{x[t], y[t]},t]