Use Taylor's Theorem to Find the First Six Terms of the Series

question 30

Multiple Choice

Use Taylor's Theorem to find the first six terms of the series solution of $\text { of } y ^ { tt } - 2 x y = 0$ given the initial conditions $y ( 0 ) = 5$ and $y ^ {t } ( 0 ) = - 7$

• A)
• \(y = 5 - \frac { 7 } { 1 ! } x + \frac { 10 } { 3 ! } x ^ { 3 } - \frac { 28 } { 4 ! } x ^ { 4 } + \frac { 80 } { 6 ! } x ^ { 6 } - \frac { 280 } { 7 ! } x ^ { 7 } + \cdots\)
• B)\(y = 7 - \frac { 5 } { 1 ! } x + \frac { 14 } { 3 ! } x ^ { 3 } - \frac { 20 } { 4 ! } x ^ { 4 } + \frac { 112 } { 6 ! } x ^ { 6 } - \frac { 200 } { 7 ! } x ^ { 7 } + \cdots\)
• C)\(y = 7 - \frac { 5 } { 1 ! } x + \frac { 14 } { 3 ! } x ^ { 3 } - \frac { 20 } { 4 ! } x ^ { 4 } + \frac { 112 } { 7 ! } x ^ { 7 } - \frac { 175 } { 8 ! } x ^ { 8 } + \cdots\)
• D)\(y = 5 - \frac { 7 } { 1 ! } x + \frac { 10 } { 3 ! } x ^ { 3 } - \frac { 28 } { 4 ! } x ^ { 4 } + \frac { 90 } { 7 ! } x ^ { 7 } - \frac { 245 } { 8 ! } x ^ { 8 } + \cdots\)
• E)\(y = 5 - \frac { 7 } { 1 ! } x + \frac { 10 } { 2 ! } x ^ { 2 } - \frac { 28 } { 3 ! } x ^ { 3 } + \frac { 80 } { 4 ! } x ^ { 4 } - \frac { 280 } { 5 ! } x ^ { 5 } + \cdots\)

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