7.38, §3 Eigenfunctions

Substitute into the homogeneous P.D.E. :

Sturm-Liouville problem for

already in standard form, with periodic boundary conditions

Pretend we do not know the solution!

Characteristic polynomial:

Case :

Since k₁ = k₂ = 0:

Boundary conditions:

hence . No undetermined constants in eigenfunctions! Simplest is to choose A=1:

Case :

Boundary conditions:

Instead of directly trying to solve, (messy), this time let us write out the matrix of the system of equations for A and B:

Then, for a nontrivial solutions, the determinant must be nonzero:

Hence must be positive, and be one of

The system for A and B becomes:

There are two undetermined constants in the solution:

First eigenfunction:

Second eigenfunction from Gram-Schmidt:

Total:

Sturm Liouville theorem: for any function

12/06/00 0:18:42