@acer thanks sir for help. It looks bit complex for me and do not find or observe the exact values of A and B? is it possible tyo solve as simultinouly?

what is meant by sigma = -sigma, acutually, u[1] = 0 at y=  --sigma  and then u[1] = 0 at y=+sigma. 

@sand15  thanks sir, let me learn from your work.

@sand15 Sir is it possible to draw compression between this analytic solution and numeric ?

 Means our ODE have two parameter b and n.....is it possible to draw the graphs of the analytiuc solution for b in the interval [0.1..0.9] with stepsize of 0.2 and fixed n.  Now fixed the b and change the n and draw the graphs. Similarlay draw graph through any numeric method for b and n then show these graph in one graph to compare the validity of both methods? And tabular form such as ist colum have y in the range 0..1 with stepsoze of 0.1 for different values of b like [0.1,0.3,0.5,0.7,0.9] and second colum is for analytic solution and third for numeric solution and fourth for error between these two solution?

@sand15 sir I compared the problem and do jot compare the solution. One term is differ in ist oder problem. Besides this, I do not means you are wrong, sorry for this. You people are doing great work to help students and researcher.  Sir I will read whole work sheet and try learn and will apply on my problem. If i feel any problem contact you via this question.

@sand15 thank alot for your help. I am understanding your worksheet and then will apply for my own problem. I have compared the ist and second orders problem but its look not same as reported in paper. Sir, if we suppose the any numeric value of exponent then after compringshould be, zero or ist and second order problem should be same. Why it is not look same ? for refernce I am uploading the picture of paper, please have look. And sir what is the meaning or prupose of the following two commands

ansatz_order    := 3;

and 

BandK  := {b=0.2, k=1}:
solnum := dsolve(eval({HPMEq0(5), g(0)=1, D(g)(1)=0}, [b=1, k=1, p=1]), numeric);

@sand15 thanks for uplod this worksheet. I am trying but couldn't get the zero and ist order problem as reported in the pictures that I have uploaded in my question. Actually, my original ode have 'N' in exponent when i puttbis N to some numeric number I get some equations but not similar to the paper. Sir if possible then can yoy please soare some time and have look on my question and reproduce that results in my worksheet. I have uploded my maple worksheet and whole problem and solution of the paper.

@sand15 I admit your observation is ok but you can varify from all aspect that I do not know the other. I am revising the article and my friend suggested me that Maple software is good in symbolic computation or solution, so used Maple. So install Maple and search for help or code of the mentioned method on google. I found this plate form and create my account by yesterday. Luckly, I found the latest question of HPM and downloded the worksheet of your response and start replacing my problem. I even do not change the lables and other constant even I mark my constant according to the worksheet because i do know how to write in Mpale. This is all explanation from my side. I do know the other guys. If you believed then help out me.

@sand15 do not know the other guys and may be the time if asking question is same and you feel we are same...........wokring on HPM for already published paper, once reproduced the result then will apply or use for my own problem. Please help to produce results mentioned in the pictures , I have uploaded..

Note: I have follow your response to question asked by some oine few days ago.