Hey I'm trying to answer the exact same problem for homework and am really confused..I get as far as;
But then I'm not sure what to do - I can see I'm going to get different results if u >3, <3, =3, but I don't know how to put that into an overall answer... at what point should I be using the boundary conditions?
Also, the 2nd part of the question asks me to expand a hypothetical f(x) in terms of the eigenfunctions - I assume in the format where Ck are the sturm-louiville coefficients. But how can I do this if the format of Yk changes depending on whether u is > or < 3? Confused... :-s
. That corresponds to your three cases.
Cases 1: then
Now use your BC's.
Two equations for the two unknowns . The first says whereas the second
so which gives A=0 leading to the trival solution . Similarly for the second case. It's the third case that's interesting. You try now.