In the Previous Three Problems, the Solution of the Original $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )$

question 17

Multiple Choice

In the previous three problems, the solution of the original problem is

• A) \(u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )\) , where \(c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / \sinh ( - n \pi H / L )\)
• B) \(u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )\) , where \(c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )\)
• C) \(u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \cosh ( n \pi ( y - H ) / L )\) , where \(c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )\)
• D) \(u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \cosh \left( ( n \pi / L ) ^ { 2 } ( y - H ) \right)\) , where \(c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )\)
• E) \(u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) \sinh \left( ( n \pi / L ) ^ { 2 } ( y - H ) \right)\) , where \(c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / \sinh ( - n \pi H / L )\)

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Definitions:

Two-Dimensional Graph

A visualization tool showing the relationship between two variables on a plane, with one variable along the x-axis and the other along the y-axis.

Overallocated Resources

Refers to a situation where more resources are being used in the production of a good or service than what is considered efficient, leading to wastefulness or reduced economic welfare.

Marginal Benefits

The additional satisfaction or value that one receives from consuming or producing one more unit of a good or service.

Production Possibilities

A curve or frontier that shows various combinations of the amounts of two goods or services that can be produced within the limits of available resources and technology.