if i change to what you say,

> dsol := dsolve(`union`(ex1, ic), numeric);
proc(x_rkf45)  ...  end;

How to show dsol? And how to odeplot this? And how to phaseportrait this?

> odeplot(dsol, [t, f(t)], 0 .. 100);
Error, (in plots/odeplot) curve is not fully specified in terms of the ODE solution, found additional unknowns {f(t)}

> phaseportrait(ex1, [f1(t), f2(t), f3(t), f4(t)], t = [[f1(0) = 0, f2(0) = 0, f3(0) = 0, f4(0) = 0]], [f1 = 0 .. 5, f2 = 0 .. 5, f3 = 0 .. 5, f4 = 0 .. 5]);
Error, (in DEtools/phaseportrait) invalid range for independent variable

Moreover, how to phaseportrait it , parameters are quite complicated

as i can not edit , i discover an typing error

+ 2*(-G*M/(r*(-r*c^2+2*G*M)))*Diff(f(t), t1)*Diff(f2(t), t) = 0,

change to

+ 2*(-G*M/(r*(-r*c^2+2*G*M)))*Diff(f1(t), t)*Diff(f2(t), t) = 0,
but still have error

> dsol := dsolve(`union`(ex1, ic), numeric);
Error, (in DEtools/convertsys) unable to convert to an explicit first-order system

Last equation has no error, Diff(f3(t), t$2) mean differentiate twice

Diff(f3(t), t$2)
+ 2*(1/r)*Diff(f2(t), t)*Diff(f4(t), t)
+ 2*(cos(theta)/sin(theta))*Diff(f3(t), t)*Diff(f4(t), t) = 0

It works, is it the graph represent the motion on submanifold? How to see resnum? can i interpolate resnum? if so, does the interpolation equation represent the solution?

It works, is it the graph represent the motion on submanifold? How to see resnum? can i interpolate resnum? if so, does the interpolation equation represent the solution?

How to wedge product dw := -w ^ w ?

How to wedge product dw := -w ^ w ?