DSolve[{2*x[t] + y[t] + Derivative[1][x][t] + Derivative[1][y][t] == E^(2*t) + t, -x[t] + 3*y[t] + Derivative[1][x][t] + Derivative[1][y][t] == -1 + E^t},{x[t], y[t]},t]