DSolve[{Derivative[1][x][t] == a*x[t] + g*y[t] + beta*z[t], Derivative[1][y][t] == g*x[t] + b*y[t] + alpha*z[t], Derivative[1][z][t] == beta*x[t] + alpha*y[t] + c*z[t]},{x[t], y[t], z[t]},t]