In separating variables, using, $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ , in the equation $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 2 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } + \frac { \cot \theta } { r ^ { 2 } } \frac { \partial u } { \partial \theta } = 0$ , the resulting equation for $R$ and is
A) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda R = 0$
B) $r ^ { 2 } R ^ { \prime \prime } - 2 r R ^ { \prime } - \lambda R = 0$
C) $r R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda r R = 0$
D) $r R ^ { \prime \prime } + 2 R ^ { \prime } - \lambda r R = 0$
E) $r ^ { 2 } R ^ { \prime \prime } - 2 r R ^ { \prime } - \lambda r R = 0$

Question

Quizplus · Accepted Answer

Substituting $u(r, \theta) = R(r)\Theta(\theta)$ into the given partial differential equation and separating variables, the equation for $R(r)$ involves terms with $r^2 R''$, $2rR'$, and a separation constant $-\lambda$, leading to $r^2 R'' + 2rR' - \lambda R = 0$.

In Separating Variables, Using u(r,θ)=R(r)Θ(θ)u ( r , \theta ) = R ( r ) \Theta ( \theta )u(r,θ)=R(r)Θ(θ)

In Separating Variables, Using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$