In the previous problem, after separating variables using $u ( r , t ) = R ( r ) T ( t )$ . the resulting problems are
A) $r R ^ { \prime \prime } + R ^ { \prime } + r \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } - \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$
B) $r R ^ { \prime \prime } + R ^ { \prime } + r \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$
C) $r R ^ { \prime \prime } + R ^ { \prime } - r \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$
D) $r R ^ { \prime \prime } + R ^ { \prime } + \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$
E) $r R ^ { \prime \prime } + R ^ { \prime } + \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } - \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$

Question

Quizplus · Accepted Answer

The correct separation of variables for a problem involving a radial function $R(r)$ and a time-dependent function $T(t)$ typically results in a spatial part that includes the term $r R'' + R' + r\lambda R = 0$ (or a variation thereof) and a temporal part that is a second-order differential equation. The correct temporal part should have a positive sign before $\lambda$ because it represents a harmonic oscillator equation, $T'' + \lambda T = 0$, which describes oscillations or waves in time. The conditions given for $R(r)$ being bounded at $r=0$ and $R(2)=0$ are consistent with boundary conditions for a physical problem defined in a radial domain.

In the Previous Problem, After Separating Variables Using u(r,t)=R(r)T(t)u ( r , t ) = R ( r ) T ( t )u(r,t)=R(r)T(t)

In the Previous Problem, After Separating Variables Using $u ( r , t ) = R ( r ) T ( t )$