@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 

@dharr  i wish i have a good book for study this subject, i have  wronge paper in front of me  and i have to work on it, i will do the rest thank a lot .

@dharr We couldn't have done it without you ,becuase of that i have to check all result really there is a lot of this kind i have to check all of them,again i did another question like that but more complicated, i don't know why my work didn't give me answer , i ask too much i am shaming,  you are amazing 

@dharr i did everything correctly but not get zero
T1.mw

@dharr it been 2 day i am stuck at this , thanks a lot

@dharr please if you check this send me file is i can't reach zero