The Solutions of a Regular Sturm-Liouville Problem $\left( \left( r y ^ { \prime } \right) ^ { \prime } + ( \lambda p + q ) y = 0 , y ( a ) = 0 , y ( b ) = 0 \right)$

Multiple Choice

The solutions of a regular Sturm-Liouville problem $\left( \left( r y ^ { \prime } \right) ^ { \prime } + ( \lambda p + q ) y = 0 , y ( a ) = 0 , y ( b ) = 0 \right)$ have which of the following properties?

A) There exists an infinite number of real eigenvalues.
B) The eigenvalues are orthogonal on $[ a , b ]$ .
C) For each eigenvalue, there is only one eigenfunction (except for non-zero constant multiples) .
D) Eigenfunctions corresponding to different eigenvalues are linearly independent.
E) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function $r ( x )$ on the interval $[ a , b ]$ .