In the Previous Problem, a Series Solution Corresponding to the Indicial

question 21

Multiple Choice

In the previous problem, a series solution corresponding to the indicial root $r = 1 / 2$ is $y = x ^ { 1 / 2 } \left\{ 1 + \sum _ { k = 1 } ^ { \infty } c _ { k } x ^ { k } \right\}$ , where

• A) \(c _ { k } = ( - 2 ) ^ { k } / [ k \mid 3 \cdot 5 \cdot 7 \cdots ( 2 k - 3 ) ]\)
• B) \(c _ { k } = ( - 2 ) ^ { k } / [ k \mid 1 \cdot 3 \cdot 5 \cdots ( 2 k - 3 ) ]\)
• C) \(c _ { k } = - 2 ^ { k } / [ k \mid 5 \cdot 7 \cdot 9 \cdots ( 2 k + 1 ) ]\)
• D) \(c _ { k } = ( - 2 ) ^ { k } / [ k ! ( 2 k + 3 ) ! ]\)
• E) \(c _ { k } = ( - 2 ) ^ { k } / [ k \mid 5 \cdot 7 \cdot 9 \cdots ( 2 k + 3 ) ]\)

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Definitions:

Alkaline Solution

A solution with a pH value greater than 7, indicating it has fewer hydrogen ions than pure water and is commonly referred to as basic.

Basic Solution

A solution that has a pH greater than 7, indicating it has a higher concentration of hydroxide ions than hydrogen ions.

Organic Molecules

Compounds primarily made of carbon atoms, often containing hydrogen, oxygen, and other elements, that are the basis of life.

Weak Base

A base that partially dissociates into its ions in an aqueous solution, resulting in a weakly alkaline solution, as opposed to a strong base which fully dissociates.