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

question 1

Multiple Choice

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

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

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Definitions:

Gross Profit

The difference between sales revenue and the cost of goods sold, before deducting overhead, payroll, taxation, and interest payments.

Ending Inventory

The aggregate value of products on hand for selling when an accounting cycle concludes.

Gross Profit

The difference between revenues and the cost of goods sold before deduction of operating expenses, interest, and taxes.

Ending Inventory

The value of goods available for sale at the end of an accounting period, calculated as the beginning inventory plus purchases minus the cost of goods sold.