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

Multiple Choice

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) $p$ , $q$ , $r$ piecewise 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) $A _ { 1 } B _ { 1 } \neq 0$
E) $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } \neq 0$