Can someone help me with this?
How do I find the eigenvalues and eigenfunctions for the Sturm-Liouville problem
;
Thanks in advance.
That equation is the same as , an "Euler type" or "equipotential" equation. The change of variable t= ln(x) changes it to an equation with constant coefficients. You should be able to find the general solution. Obviously, y(x)= 0, for all x, satifies the equation as well as y(1)= y(2)= 0. The eigenvalues are the values of u for which there are other, non-trivial, solutions. And the eigenfunctions are the non-trivial solutions.
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
The best thing to do is consider the following three cases:
. 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.