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 $[ a , b ]$ .

Question

Quizplus · Accepted Answer

The Answer of 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 $[ a , b ]$ .

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

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