@Alfred_F this method is new , it is easy but need a undrestanding, because this all solution must satisfy the ode, i hope some one undrestand and give me a good solution

@dharr @mmcdara  

Please don’t make it sound like you’re doing me a favor as a burden. I truly appreciate your efforts and the help with the code, but the way it's being said makes it seem like I did something wrong on purpose , which I definitely didn’t.
The code I shared was from an older version. I only modified it where I thought it was necessary. I’ve been working on a new equation, and since the structure was similar, I assumed it wouldn’t make much of a difference , just the equation itself was new.
I couldn’t have known that a small change would be considered so significant. At the end of the day, the core idea remains the same, and if someone is experienced in the work, they’ll easily recognize that.

betwen this file and sys there is not different just ode equation is different the idea it same 

old-orginal.mw

@Alfred_F i try to figure out but i don't have idea about this 

@dharr i am looking for make a system of coefficient of that term I mentioned before, then i solved that system for finding parameter involved in U(xi)  which make my ode zero.

@mmcdara you did your help in best shape you can, thanks you.

@mmcdara i want to do this, i want make thus equation eq0-eq6 find the parameter involving with G for get my ode zero  i want do systematically make thus eq like you did but i can't use and change your code is hard a little bit  for me

please see sys.mw

@dharr i want factoring [1/G')^i and  (G'/G)*(1/G')^i i want factoring this two term the second one is  together  in equaation G , i just write thus system in below for more undrestanding which i did it by hand one by one by copy paste in each coeffecent this is a little bit hard each time i do that i want do systematically

@dharr it is different from the other  and i think it is not apply that factoring  in here we have two multiply term in last one we have just one 

@dharr in last of each equation i write this equation come from what in fact i have to factoring   like that , i don't want factor eq i want factoring G1, in last of each equation i have coeffecent of that term 

and this is come from this series 

@dharr amazing, is not my fault sorry ,i did my best try but each time need your hand to go on....thank you Dr

@dharr it work thanks dear dctor 

@dharr i thought i can hundle other step really i did samething but is not apply for general form , i did the samething but not apply i want solve ode  but becuase the step are too long i just ask where i stuck otherwise i will upload all, like this

@dharr is nonsense is just becuase sign of case instead of this {} i write this [] , thanks dear dctor i will upadte my old code , and referesh all, thanks a lot Dr dctor david

@nm  you undrestand totally correct, maple give me one solution, so i have auxiliary differential equation which have more than 20 different solution and i write down and tested all of them and all of them are answer, becuase by this ode i want solve pde so How many this solution are be more and more i have more solution of pde, iam not working on ode like you do and i don't have enough information about that method you talking about but if exist please provide me and give me something to get for any auxiliary ode i get at least twenty different solution  that make my method more power full then a lot of author in the paper modefied their method just by adding 1 or more ode solution,  your comment are right about the other method maybe they use  the otehr method to achieve that solution ,  for example for this ode in one paper which already exist the paper if you want i can mentioned , he talk about one ode with more than 27 solution i try to test them and all of them are correct please take a look for undrestanding more, also about your question ( i am looking for generate series solution but i need to watch what exactly they did and how find out) , if other method give me at least twenty is good but if not , i have to make a generate function which contain (sin-cos-sec-csc- sinh-cosh-sech,csch-tan-tanh-cot-coth) also we can use generate function based of exponential too, also based of the parameter contain in our ode and do odetest for it if is answer take this out , iam looking for this , this is based on your experience possible?

ode-test.mw

@janhardo  thank you john this book is so ggod for my information i didn't see it before , 
if you have any other please mention for me
also there is any other book which when we have a ode then list the solution of each ode how many solution have it listed all of them i am search for something like that right now becuase a lot equation exist which when we solve by Dsolve option he just give us 1 solution and no more , i am looking for generate series solution but i need to watch what exactly they did and how find out if you have a book or lecture note please mention it