DSolve[{(x[t] - y[t])*(x[t] - z[t])*Derivative[1][x][t] == f[t], (-x[t] + y[t])*(y[t] - z[t])*Derivative[1][y][t] == f[t], (-x[t] + z[t])*(-y[t] + z[t])*Derivative[1][z][t] == f[t]},{x[t], y[t], z[t]},t]