Consider the Differential Equation $2 x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + ( 2 x - 1 ) y = 0$

question 38

Multiple Choice

Consider the differential equation $2 x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + ( 2 x - 1 ) y = 0$ The indicial equation is $2 r ^ { 2 } + r - 1 = 0$ . The recurrence relation is $c _ { k } [ 2 ( k + r ) + ( k + r - 1 ) + 3 ( k + r ) - 1 ] + 2 c _ { k - 1 } = 0$ . A series solution corresponding to the indicial root $r = - 1$ is $y = x ^ { - 1 } \left[ 1 + \sum _ { k = 1 } ^ { \infty } c _ { k } x ^ { k } \right]$ , where

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

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Definitions:

Paraphilic Disorder

A disorder in which a person’s paraphilia causes great distress, interferes with social or occupational activities, or places the person or others at risk of harm—either currently or in the past.

Hormonal Imbalance

A condition in which there is a disparity in the body’s normal levels of hormones, affecting bodily functions.

SSRIs

Selective Serotonin Reuptake Inhibitors, a class of drugs often used in the treatment of depression and anxiety by increasing serotonin levels in the brain.

Paraphilias

Unusual or atypical sexual interests or practices that may cause distress or impairment in functioning or harm to oneself or others.