```

Case 1: intermediate output, root-finding

Calculation with tol =    1.0e-04

X         Y(1)         Y(2)         Y(3)
0.00      0.50000      0.50000      0.62832
(User-supplied callback fcn, first invocation.)
(User-supplied callback g, first invocation.)
2.00      1.54931      0.40548      0.30662
4.00      1.74229      0.37433     -0.12890
6.00      1.00554      0.41731     -0.55068

Root of Y(1) = 0.0 at   7.288

Solution is  -0.00000   0.47486  -0.76011

Calculation with tol =    1.0e-05

X         Y(1)         Y(2)         Y(3)
0.00      0.50000      0.50000      0.62832
2.00      1.54933      0.40548      0.30662
4.00      1.74232      0.37433     -0.12891
6.00      1.00552      0.41731     -0.55069

Root of Y(1) = 0.0 at   7.288

Solution is  -0.00000   0.47486  -0.76010

Case 2: no intermediate output, root-finding

Calculation with tol =    1.0e-04

Root of Y(1) = 0.0 at   7.288

Solution is  -0.00000   0.47486  -0.76011

Calculation with tol =    1.0e-05

Root of Y(1) = 0.0 at   7.288

Solution is  -0.00000   0.47486  -0.76010

Case 3: intermediate output, no root-finding

Calculation with tol =    1.0e-04

X         Y(1)         Y(2)         Y(3)
0.00      0.50000      0.50000      0.62832
2.00      1.54931      0.40548      0.30662
4.00      1.74229      0.37433     -0.12890
6.00      1.00554      0.41731     -0.55068
8.00     -0.74589      0.51299     -0.85371
10.00     -3.62813      0.63325     -1.05152

Calculation with tol =    1.0e-05

X         Y(1)         Y(2)         Y(3)
0.00      0.50000      0.50000      0.62832
2.00      1.54933      0.40548      0.30662
4.00      1.74232      0.37433     -0.12891
6.00      1.00552      0.41731     -0.55069
8.00     -0.74601      0.51299     -0.85372
10.00     -3.62829      0.63326     -1.05153

Case 4: no intermediate output, no root-finding ( integrate to xend)

Calculation with tol =    1.0e-04

X         Y(1)         Y(2)         Y(3)
0.00      0.50000      0.50000      0.62832
10.00     -3.62813      0.63325     -1.05152

Calculation with tol =    1.0e-05

X         Y(1)         Y(2)         Y(3)
0.00      0.50000      0.50000      0.62832
10.00     -3.62829      0.63326     -1.05153
```