Use Power Series to Solve the Differential Equation $\left( x ^ { 2 } + 1 \right) y ^ { \prime \prime } + x y ^ { \prime } - y = 0$

question 51

Multiple Choice

Use power series to solve the differential equation.. $\left( x ^ { 2 } + 1 \right) y ^ { \prime \prime } + x y ^ { \prime } - y = 0$

• A) \(y ( x ) = c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n }\)
• B) \(y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n + 1 }\)
• C) \(y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } ( n - 2 ) ! } x ^ { 2 n }\)
• D) \(y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n }\)
• E) \(y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 2 n ) ! } { 2 ^ { 2 n + 2 } n ! ( n - 2 ) ! } x ^ { 2 n + 1 }\)

---

Definitions:

British America

Territories in the Americas that were colonized by Britain, including parts of the United States, Canada, the Caribbean, and other regions, up until the 20th century.

African Religions

The diverse spiritual practices, beliefs, and traditions indigenous to the African continent, characterized by a rich tapestry of cultural expressions, ancestral worship, and community rituals.

Stono Rebellion

was a slave uprising in 1739 in South Carolina, one of the largest and most significant slave rebellions in the colonial United States.

Slavery

A system in which individuals are owned by others, deprived of personal freedom, and compelled to perform labor or services without compensation.