Last time, I was stuck by Sturm Liouville Problem. Finding the eigenvalue.
, and differentiable.
Given the boundary condition:
and eigen-function satisfies that:
, if .
The theory states that:
- The eigenvalues
- Every eigenvalue is simple(algebraic multiplicity is one).
- There exists an orthogonal basis mentioned above.
Due to the inner product, we can easily solve one case, if we can do the integral by changing variables. If
That is , or we can relax the situation by .
It is easy to see if we find another variable , then
But for other cases, it won’t be easy to find out the eigenvalues easily. However, we ruled out the trivial cases that .