To my knowledge, it can be use in a square domain only(i.e. x1 in [a,b] and x2 in [c,d]).
Non-square domain is usually transformed into square domain by the change of variable, in order to separate the variables.
I get the same problem before when the first time I saw this formula.
Why f(y)=\int_y^b b_m(s) sinh(s)ds is not trival..
It is because the term f(y)=\int_0^y b_m(s) sinh(b-s)ds has been combined with the integration constant(or from the constant of homogenous solution) which equals to...
The dummy variable does not make any difference.I think your calculation is the same as that on your book.But the only difference is: that one on your book has already put the boundary conditions "u=0 at y=0 and b" into the two intergration constants arisen from the two integrations.
By I think...