" in the end - many more things must be checked still I feel". my concern is that this problem is not well suited to NDSolve except with deeper knowledge of method used and actual error. you asked about only use of Inverse and whether NDSolve would accept a symbolic inverse or would demand a real inverse.

tenenbaum, p. 562, series solutions to nonlinear ODE of order greater than one - this method requires that the usable convergence interval, I: |x-x0| < min(r, r/((n+2)M)) be found and respected

Thank you again for all the help. I went back and revised the initial condition to something that might work and use your latest code. Also be aware that I'm looking for only a Real solution. The problem with this solution is that the Series expansion works on smooth function, but this is a warm-up problem to a more difficult problem where the S(x) inverse is extremely stiff, and steep. I attached the revised file with a "better" more physical boundary condition with your modification. Please let me know if you have other ideas for a stiff one? It seems to me that that problem of using a function results from NDSolve to solve a secondary ODE equation should be used by others.

Attachments:

R power of 2new ...nb