DSolve[{Derivative[1][x][t] + Derivative[1][y][t] == x[t]*y[t], Derivative[1][y][t] + Derivative[1][z][t] == y[t]*z[t], Derivative[1][x][t] + Derivative[1][z][t] == x[t]*z[t]},{x[t], y[t], z[t]},t]