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 \([ a , b ]\) .

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