@tomleslie 

Excuse me, I mean, how to find these terms without using Maple (so I'm trying to understand how to get terms purely mathematical, the theoretical idea for evaluating it)...

@tomleslie

It's clear, but I'm interested what how to analyze this psi(x) without maple, maybe you know some works of literature, references, i.e., you yourself should know how to get these coeffs? 

Do I understand correctly that this 4*l^2*sin(n*Pi/2) and  l^2((-1^n)-1) the same thing?

@tomleslie 

Sorry, but it is by no means my desire, to give the solution in such detail!

@tomleslie 


Thank you for the much-needed help! Just let me ask you one more question! How is it all analytic to make a change of variable x and get an interval (-l/2;l/2) in order to examine the function for even and odd parity, somehow the maple calculates these coefficients using the commands you gave?

@tomleslie 

Thanks for this worksheet file and solution to the equation! I just need to get the result step by step, so I separate the variables, solve two different ODEs, but at the moment of calculating the expansion coefficients psi (x), there was a problem! Could you tell me, maybe how I can still consider such a case in maple and implement it: perhaps I should replace x so that the interval becomes (-l / 2; l / 2), and think about when the integrand will be even / odd ...

@mmcdara 

I think it is the wrong result, rather, even I don't understand. Maybe should make a replacement x so that the interval becomes (-l/2;l/2) and consider when the integrand will be even / odd. Could you explain to me, how to implement it in Maple? Sorry for my ignorance, but I need the right analytical solution, I implement the numerical solving, but I need to compare it with the analytical solution...

@mmcdara 

Thanks a lot! Now I can write the general solution! 

@mmcdara 

It's working, but how to integrate this in maple:

I can't integrate this expression without CAS, and how I could get the result with the normal form?