Using Power Series Methods, the Solution Of $2 x y ^ { \prime \prime } + y ^ { \prime } + 2 y = 0$

question 45

Multiple Choice

Using power series methods, the solution of $2 x y ^ { \prime \prime } + y ^ { \prime } + 2 y = 0$ is

• A) \(\begin{array} { l }
• y = c _ { 0 } \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) + \\
• c _ { 1 } x ^ { 1 / 2 } \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) )
• \end{array}\)
• B) \(\begin{array} { l }
• y = c _ { 0 } \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) + \\
• c _ { 1 } x ^ { 1 / 2 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]
• \end{array}\)
• C) \(\begin{array} { l }
• y = c _ { 0 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) \right] + \\
• c _ { 1 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]
• \end{array}\)
• D) \(\begin{array} { l }
• y = c _ { 0 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) \right] + \\
• c _ { 1 } x ^ { 1 / 2 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]
• \end{array}\)
• E) \(\begin{array} { l }
• y = \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) \right] + \\
• x ^ { 1 / 2 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]
• \end{array}\)

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Definitions:

Required Reserves

The minimum amount of funds that a bank must hold in reserve against deposit liabilities, as mandated by central banking authorities.

Reserve Bank

A central banking institution responsible for overseeing the monetary policy, issuing currency, and regulating the banking system of a country.

Required Reserve Ratio

The fraction of deposits that a bank is required by regulation to hold in reserve and not lend out, aiming to ensure bank liquidity and stability.

Money Supply

The total amount of monetary assets available in an economy at any specific time, including cash, coins, and balances held in checking and savings accounts.