DSolve[{Derivative[1][x][t] == x[t]^2/2 - y[t]/24, Derivative[1][y][t] == 2*x[t]*y[t] - 3*z[t], Derivative[1][z][t] == -y[t]^2/6 + 3*x[t]*z[t]},{x[t], y[t], z[t]},t]