A Power Series Solution About $x = 0$ Of the Differential Equation

question 23

Multiple Choice

A power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } + y = 0$ is

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

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Definitions:

Proposition

A statement that expresses a judgement or opinion which can be either true or false.

Modus Ponens

A logical argument form where if a conditional statement ("if p then q") and its antecedent (p) are true, then the consequent (q) is also true.

Antecedent

In logic and grammar, the antecedent is the preceding clause, phrase, or word to which another word, particularly a pronoun, refers back.

Pure Hypothetical

A speculative statement or proposition that is considered in abstraction from any particular instance or factual situation.