Solve the Initial-Value Problem $$( 1 + \cos x ) \frac { d y } { d x } = \left( 6 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$$

question 52

Multiple Choice

Solve the initial-value problem. $$( 1 + \cos x ) \frac { d y } { d x } = \left( 6 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$$

• A) \(y = \frac { \ln ( 6 + \cos x ) } { \ln ( 6 - \cos x ) }\)
• B)
• \(y = \ln \left( \frac { 1 } { 6 } \left( \sec ^ { 12 } \left( \frac { x } { 2 } \right) - 1 \right) \right.\)
• C) \(y = \ln ( 6 - \cos x ) ( 6 + \cos x )\)
• D)
• \(y = \ln \left( \frac { 1 } { 6 } \left( 7 \sec ^ { 12 } \left( \frac { x } { 2 } \right) - 1 \right) \right.\)
• E)
• \(y = \ln \left( 6 \sec \left( \frac { x } { 2 } \right) - 1 \right)\)

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Definitions:

Public Goods

Goods that are non-excludable and non-rivalrous, which means that one person's use of the good does not reduce its availability to others, and no one can be effectively excluded from using the good.

Private Philanthropy

Financial contributions or aid given by individuals or private organizations to support various causes and non-profit organizations.

Free-rider Problem

A situation where individuals benefit from resources, goods, or services without paying for them, leading to underprovision of those public goods.

Nonrivalry

A property of goods where one person's consumption does not prevent consumption by others, typically relevant in the context of public goods.