@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

@dharr  @acer  my Question is a little bet more but i though if i can do this step the rest i will do by myself but i don't know How to replacing in next other really  iam sorry   i can't replace even i have this amazing code is a little bit confusing , this is first time this happen

A.mw

@dharr in fact i am looking forward to find a generation function and use a loop for test that generation function for finding solution for ode, if possible i am sure it is possible even if maple can't find them i don't have knowledge to write down the code but i have idea , becuase i saw a lot of ode which no any mathematical program can find but they tested manually by hand but if we do by generation function and take out that function which make ode zero, i will post a new question about this topic ofcurse we can start by small ode and small generation function 

@dharr all of that function i try more than 3 kind all of them make a problem and when i found parameter the parameter set of parameter are so few and didn't give me answer for this equation i think this way is not usefull for getting solution i will look for another equation, thank you dear Dr

@dharr i found he did this part 

@dharr in fact, this is first time i solve this  new equation, the other equation the monomial of them is not to much but in here is more than 10 , i learned this trick from another paper, which have a lot term he just say take coeficcent of thus, i say maybe this trick give me answer , also my other problem of mine i don't want find all parameter just thus which contain in (f), and when i do that it not found and just runing , i am looking for some pretty solution of this pde and really i don't know the true idea behind of this exactly,i want choose thus parameter which make my PDE zero,  also in file have two H function please delete the second one which have one derivative in right hand side

S11-Dr.D.mw

@dharr thank you so much ,

@dharr if i am  automatically not get explicit answer i have to make them which one of them which this take a time, no problem thanks so much