@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?