The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is a regular Sturm-Liouville problem under certain conditions, including Select all that apply. A) $p$ , $q$ , $r$ are continuous on $[ a , b ]$ B) $r ( x ) 0$ and $p ( x )

Question

The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is a regular Sturm-Liouville problem under certain conditions, including Select all that apply. A) $p$ , $q$ , $r$ are continuous on $[ a , b ]$ B) $r ( x ) > 0$ and $p ( x ) < 0$ on $[ a , b ]$ C) $r ( x ) < 0$ and $p ( x ) > 0$ on $[ a , b ]$ D) $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } \neq 0$ E) $A _ { 1 } A _ { 2 } \neq 0$

Quizplus · Accepted Answer

For a regular Sturm-Liouville problem, $p$, $q$, and $r$ must be continuous on $[a, b]$, and $r(x) > 0$ on $[a, b]$. Additionally, the boundary conditions must be such that $A_1^2 + B_1^2 \neq 0$ and $A_2^2 + B_2^2 \neq 0$.

The Problem Is a Regular Sturm-Liouville Problem Under Certain Conditions, Including Select