Consider the Problem With Boundary Conditions $u ( r , 0 ) = 0$

Consider the problem $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0$ with boundary conditions $u ( r , 0 ) = 0$ , $u ( r , \pi ) = 0 , u ( 1 , \theta ) = f ( \theta )$ . Separate variables using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ . The resulting problems for $R \text { and } \Theta$ are

A) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , R ( 0 ) = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$
B) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$
C) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , R ( 0 ) = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$
D) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , R ( 0 ) \text { is bounded, } \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$
E) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$