DSolve[{11*x[t] + 31*y[t] + 4*Derivative[1][x][t] + 9*Derivative[1][y][t] == E^t, 8*x[t] + 24*y[t] + 3*Derivative[1][x][t] + 7*Derivative[1][y][t] == E^(2*t)},{x[t], y[t]},t]