@Preben Alsholm 

thanks ...why behaviour in second branch graph is different with attached graph?

very thanks

@Preben Alsholm 

thanks...yes exactly.the second branch of omega2 would correspond to a solution where s(x) had one zero between 0 and 1, not two

Also you say correct about''You used the same approximate solution for the first omega2 branch, which actually corresponds to solutions with no zeros for s(x) in the open interval 0..1''

thanks... yes result is correct, but why g(x)=0  and  s(x) = 0.001*x in AP0??..

also vertical axis should multiplication in ''L''  i.e. (S1*L)

plot(Sigma/(L*sqrt(m/E_a)), S1*L, labels = [sigma, s(1)])

thanks...

@torabi 

please answer to me..thanks

@Preben Alsholm 

Thanks a lot Dear prof. Preben Alsholm....

For drawing second result (omega2(1)=0.144201112054868), error ''"initial Newton iteration is not converging" is occure?

Also behaviour of this graph is different with origin.please see it and help to me again....

thanks

omega=30_2.mw

These results must be achieved!!

@torabi 

FOR ME NOT ANSWER TOO!!!!

@tomleslie 

@Preben Alsholm 

hello..may i ask you for reply to me?

http://www.mapleprimes.com/questions/212370-About--Nonlinear-ODE-System-Again#comment229610

thanks alot

@Preben Alsholm 

hello mr prof. Alsholm.thanks alot for former your attention.

i have a mistake before and now whole results are correct and i apology from you.

but i have only one question.how i can draw a graph (s(x))(1) versus sigma?

result should be same with this figure in the following.

only vertical axe ((s(x))(1)) should be multiplication in coefficient [L].

omega=30.mw

@Preben Alsholm 

THANKS...i check it again...

but in final answer for example (R[4, 48])why X and Z is not apply on matrix and shown in final result?

thanks alot

@Preben Alsholm 

hi ...Hello Dear prof   Preben Alsholm 

thanks for helping me about former question.

another my question is about problem that there is different 

differential operator such as

before only exist 

i dont know how change following program?

please help me..

maple code also attached ...

maple3.mw

thanks alot

@Preben Alsholm 

may i ask you, for spending time on  the pdf-file i uploaded, before

thanks alot

@one man 

I dont need to ROOTOF!!!

@Preben Alsholm 

hi Mr Preben Alsholm ..thanks alot for good comment about this probem.
I put a lot of time on this issue and much thought about it.your answer is correct ....but i dont understand for amount that must to be specialty to parameter b to get the answer in manuscript?

according to your Commented ''Small amplitudes''

 how i can specialty amount to parameter b, that  the values found for omega2 are not constant?how i choose this value?

important issue for me is specialty amount to parameter b?

how i help from graphs?may i ask you to create the graph for this equations same pendulum ?

thanks a lot

@Preben Alsholm 

Thanks.please say your comment about this problem 

@Preben Alsholm 

thanks...

yes i compare the result with a manuscript.the table of this manuscript attached.

only use the realation    '''sqrt(omega2)/(L*sqrt(m/E_a))''' for reach to the result of that paper.

by changing amount b or approx sol for omega2 i can reach to that result but i worry about a new equations that i do not have that answer for checking.....you say correct that

omega2 is closely related to the period of the beam which depends on the amplitude in the nonlinear case.

three_result_for_omega=0_.mw

three_result_for_omega=30.mw