Consider the Equation $u _ { x } + u _ { y y } = 0$

question 39

Multiple Choice

Consider the equation $u _ { x } + u _ { y y } = 0$ with conditions $u ( 0 , y ) = 0 , u = ( L , y ) = 0 , u ( x , 0 ) = f ( x ) , u ( x , H ) = 0$ . When separating variables with $u ( x , y ) = X ( x ) Y ( y )$ , the resulting problems for $X , Y$ are

• A) \(X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( H ) = 0\)
• B) \(X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( H ) = 0\)
• C) \(X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 0 ) = 0\)
• D) \(X ^ { \prime \prime } - \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( H ) = f ( 0 )\)
• E) \(X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( 0 ) = f ( x )\)

---

Definitions:

Explicit Costs

Direct, out-of-pocket payments for costs of production, such as wages, rent, and materials.

Entrepreneur's Potential Earnings

The forecasted income an entrepreneur can expect from their business venture, taking into account risks and opportunities.

Annual Lease

A contract for the rental of property, typically real estate, that lasts for one year.

Payments to Workers

Payments to workers include salaries, wages, benefits, and other compensation provided to employees in exchange for their labor.