@janhardo  is so good but need a little bit more compact, in final we need that u[1] and u[0] in function term not symbolic which we find one by one of them in each step seperatly u[0] is clear is initial condition with RHS of function when we take laplace inverse, but for u[1] for this equation we just have nonlinear function at RHS so, we calculate like that u[1]=invlaplace(-1/s^2*A[0]+1/s^2B[0]) this is will be u[1] for finding u[2] we have same procces u[2]=invlaplace(-1/s^2*A[1]+1/s^2B[1]) then if we calculate the final result we get a function if we substitute in our PDE must satisfy i calculate 3 of them but in fact it is go to infinity, for any other question and more undrestanding of this  topic please tell me i can help with full detail 

@janhardo  you have talent to make a good procedure i don't have that yet, i know each step to make procedure but i can't because of coding your procedure is good but have some issue like in final results not substitute u[0] u[1] and so on maybe @acer , @nm , @dharr , @Kitonum , @mmcdara , can give us a hand ofcurse they will if they have a time .

@janhardo I am looking for solved whole example by procedure and I think no one can do that 

@acer in fact i don't know anything in front you, your code are so cool, if you have any lecture note of yourself please provide for me, i wanna learn slowly slowly i want teach maple in university

@janhardo ................................

@Alfred_F  this ode equation is coming out from PDE equation have three step this is second step which i must find the solution of ode by the exist method but in some paper they find solution of this auxiliary equation and then by one case they can find 20 or even more solution so i want know how to do that then i can do that too it will be so easier if for each auxiliary equation can do that and the number solution will be increase by this we can modified method  and do sub equation methods....

@acer you did a great job but

in your case why calculate seperatly not all our parameter toghether map must be [4]? and a[0] if zero is ok but each parameter of ODE equation and in series solution assumption a[1] must not be zero which multiply by G(xi) ! in paper assumption he get 15 and more  how he achived them he must use limit i think he used limit for auxiliary equation

@janhardo ..................................................

@nm  this is your last answer? How if we don't have data like that i want find more we can finded by changing the limit or not? Like you did if we say m->1 left say m->2 right what will happen?

@janhardo I don't know why they did that 

@nm I don't know why they delete it

@janhardo  your code not run and please upload the mw file
and how if we don't have data we must able to find more solution we must looking for finding them by assumption of limiting...i don't know how but i saw a lot they did like that,

this problem is hard one really and long pde equation is (1+1) dimensional (concatenation model used for polarization-preserving fibers)

@nm  How we can add more parameter  by limit or anything else becuase there is a lot method used this technique if i can find for example at lease 100 parameter whcih give me solution this is a big achievment, i get idea from your comment so we can replace the parameter by another parameter by get limitng the result will change or not?  for example in first row m->1 (left) we can say m->2 right, what will happen to solution it will change or it will be error, how we try this? we must find thus solution even if we don't have thus data and find more solution?

@nm i will add another one question of this kind  topic again with new idea please help me in next question, i will mention you.

@janhardo  nevermind i got idea i was wrong in third part