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

question 5

Multiple Choice

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) \(r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0\)

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Definitions:

Conference Room

A room provided for singular events such as business conferences and meetings, typically equipped with audiovisual and telecommunication setups.

Memory Loss

The inability to remember information or events that one would normally be able to recall, often as a result of factors like aging, disease, or trauma.

Echoic Sensory Memory

A component of sensory memory that specifically holds auditory information, lasting for a few seconds after the initial stimulus.

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The visual component of sensory memory, allowing the mind to retain an image of a visual stimulus for a brief period after the actual stimulus is gone.