understood and thank you for your time and affort Ezra

Please disregard the previous file i send u. I noticed that i was doing something wrong, the ODE equation took the time as the series expansion of of the inverse and should be h. Also my new boundary condition was too large. But what i can see that there still an issue solving the secondry ODE with time

Attachments:

R power of 2new ...nb

I've found a problem however when checking my work: I cannot get

RR = InverseFunction[(Normal[Series[S1[x], {x, 0, 5}]] /. x -> #) &]

To "Inverse again back to S1" (not at the moment) - and I suspect that it may mean RR is "not usable". I found the below to be more believable...

RR = InverseSeries[Series[S1[x], {x, 0, 5}]];
(* the above re-inverses back to S1 in a "same plot" manner *)
G[t_] := g'[t] - ((Normal[RR] //. x -> #) &[t])/6 + 2*g[t]^2
NDSolve:  At t == 1.1`, step size is effectively zero; singularity or stiff ?
InterpolatingFunction[ . , Domain{{1.1, 1.1}}, scalar ]

I'm sorry I didn't check my steps more thoroughly the first time (i checked them but too briefly). But in the end - many more things must be checked still I feel, before the answer is usable (such as stiffness NDSolve mentioned).

Thank you, it is working. Should I be worried about this msg that I got when I run the new code? Inverse functions are being used. Values may be lost for multivalued inverses. Ezra