In the Previous Three Problems, the Solution of the Original $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$

question 16

Multiple Choice

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

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

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Definitions:

Mucus

A thick, slippery substance produced by glands in the lining of the body's passageways, such as the respiratory, digestive, and reproductive tracts, serving as a protective lubricant.

Distal Airways

The segments of the respiratory system that are furthest from the mouth, including the smaller bronchioles and alveoli, crucial for gas exchange.

Central Airways

The major air passages leading from the trachea to the lungs, facilitating the movement of air in and out of the lungs.

Pleural Effusion

The accumulation of excess fluid between the layers of the pleura outside the lungs.