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

question 43

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 \([ a , b ]\) .

---

Definitions:

Scent of Honey

An aroma associated with honey, often used in products to evoke feelings of comfort and sweetness.

Scented Deodorant

A personal hygiene product designed to prevent or mask body odor, infused with fragrances to provide a pleasant smell.

The Endowment Effect

A cognitive bias where people ascribe more value to things merely because they own them, compared to if they did not.

Sensory Marketing

Marketing strategies that target consumers' senses and emotions to create a deeper connection with the product or brand.