@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 

@dharr i try to upload last night but mapleprime didn't open, sorry for that, i try to found a solved problem before which my superviser solved but didn't tell me the trick of substitution i asked but didn't answer me, i know idea but i am not sure %100 percent about results, so in this paper i solved, i want if possible and have time for sure, take a look and if i did mistake or did something wrong please tell me or fixed the issue , becuase no one see what you see even if i see it too, your code again far amazing  and you did a preety thing, i will work on yours too , i solved this problem 
i have three new method but all of them have substitution of diff(G(xi),xi)^2 , if i do one of them correctly finding other will be easy , and there is some code which i needed i want a code of print the G(xi)^i coeffiecent which i have to know which coeffiecent is belong to which G(xi)^i, i hope this is be better for undrestanding

Dr.D-feedback.mw

@dharr  i did something new i am sure you will undrestand here by this 
in my expresion the squar is apear which not make my G(xi)^i found i did something wrong in substitution but i can't determine 

Dr.D.mw

@dharr as shown in graph maybe letter are different,

you have this series Eq(6) which in my file is F and it is equal to ode solution  U(xi), i have to take derivative from eq(6) which is diff(U(xi),xi), if watch the equation after taking derivative you will see we have substitution which we can replace diff(G(xi),xi) after replacing  we have to find diff(U(xi),xi$2) becuase our ode contain U,U', and U'', but in my equation is just U,U'',

extra note: when we do some extra thing with eq(5) like diff(G(xi),xi)=sqrt(r^2*G(xi)^2*(a + b*G(xi) + l*G(xi)^2)) if take another derivatve is is clearly  apear which contain diff(G(xi),xi) in denomenatore then it have in numerator then they will canel then we substitute the U,U',U'' then we can find system of equation based on G(xi) power the coeficent of G(xi)^i=0 they will be our system to solve the parameter contain in equation 6, if don't undrestand let me know