In the Problem , Separate Variables, Using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$

In the problem $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 2 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } + \frac { \cot \theta } { r ^ { 2 } } \frac { \partial u } { \partial \theta } = 0$ , separate variables, using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ . The resulting problems for $R$ and $\Theta$ are

A) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } + \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .
B) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .
C) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } - \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .
D) $r ^ { 2 } R ^ { \prime \prime } - 2 r R ^ { \prime } - \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .
E) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } - \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .