@mmcdara  it will be totlay correct  if instead of sigma we write sigma=A[1]^2-A[2]^2 will be true but don't know where sigma come maybe it is integral constant of solution after simplify and factoring.
also we have three type for lambda >0 and lambda=0 if you do them too it will be a big help for me becuase it is your talent i can't go jump in your code i try to change to lambda<0 but  i failed, also i want solve the whole paper of yesterday when you done and told me trick of getting (G'/G)^2 =....., if you help me at there it will be a big help becuase i stuck at factoring and isolating (1/G)^i and (1/G)^i*diff(G(xi),xi)/G also instead each (G'/G)^2 we must do substitute for each case seperatly all the same if we do one case other will be easy,i hope help me at there really i am stcuk at there,  also i am intresting in your coding is like you know what you doing to be a perfect, your help is emazing for my job.

this is other case at same paper

@acer it is totaly different from this question maybe picture are  same but demand of the question is different

@mmcdara  i must write the equation on top  picture but i write the other ode equation if you want you can change the ode to that one on top with first term s[1]w''-(mu[3]+s[1]... =0 like that or if you want i can send you pdf file if i have your email for more detail

@mmcdara i can't see any file with my maple version and i fixed your version code by simplify denominator of eq_9 i get the exact equation .

if you have time watch this file and if have any idea tell me i will do i need find thus equation in tope for my new ODE

i hope you help me this time too just for  factoring

find_parameter.mw

@mmcdara first sorry for my bad for sending a lot of question, secondly your code is totally true i can see that but the code for version of maple is not work some time have the same problem  i don't know why ...
nevermind if you can solve the equation and what code where be used for substitute if you have time it will be so good i am stuck really i know what they done in paper but for my new paper i must have a good code and if have any question i am here thank you so much

@janhardo this is a paper,

Various optical solutions for time-fractional Fokas system arises in monomode optical fibers

[full PDF link removed by moderator]

@mmcdara i watch in your uploade is correct but when i run have problem is like that

@mmcdara 6533 the denominator of eq 9 is not the same of equation  11 and 13 have the same code but outcome is different  just (1/G)^2 have a little problem

@mmcdara  i just intrested in your code my code are a little bit ugly
can you substitute eq(6) in eq(5) and also when take derivative of eq(6) must be replace G''=-sigma*G(eta)+v eq(6) and when we substite must be factorize and equal coificent to zero but we must substitute our equation like (eq9),eq(11)and eq(13)
by collecting term is  hard for substitute i don't know there is another better way for factoring and replacing

you already did the hard part of this if you do this part i will be very greatfull

@mmcdara it is wonderfull and awesome and with your explanation i hope  undrestand full subject

@Thomas Richard  it is  PDE by a lot of step i reach to this, this solution is satify for ode wothout 1/n but when we substitute to pde for test is equal to zero or not i must add power 1/n becuase the transformation i did a lot of this kind equation but i don't know what is mistak in here

@janhardo this is emazing, thank you for making rich the subject

@janhardo  thank you so much this code is emazing too

@Carl Love is so pretty emazing code

@dharr  thanks dear Dr are you best of the best