Taylor's Theorem to find the first four terms of the series solution of $y ^ {t } - 2 x y = 0$ given the initial condition $y ( 0 ) = 1$ and use it to calculate $y ( 3 )$. Round your answer to three decimal places wherever applicable.
A) $y = 1 + \frac { 2 } { 2 ! } x + \frac { 12 } { 4 ! } x ^ { 2 } + \frac { 120 } { 6 ! } x ^ { 3 } + \cdots ; y ( 3 ) \approx 13.000$
B)$y = 1 + \frac { 1 } { 2 ! } x ^ { 2 } + \frac { 6 } { 4 ! } x ^ { 4 } + \frac { 60 } { 6 ! } x ^ { 6 } + \cdots , y ( 3 ) \approx 86.500$
C)$y = 1 + \frac { 2 } { 1 ! } x ^ { 2 } + \frac { 12 } { 2 ! } x ^ { 4 } + \frac { 120 } { 3 ! } x ^ { 6 } + \cdots , y ( 3 ) \approx 15,085.000$
D) $y = 1 + \frac { 2 } { 2 ! } x ^ { 2 } + \frac { 12 } { 4 ! } x ^ { 4 } + \frac { 120 } { 6 ! } x ^ { 6 } + \cdots ; y ( 3 ) \approx 172.000$
E) $y = 1 + \frac { 2 } { 1 ! } x + \frac { 12 } { 2 ! } x ^ { 2 } + \frac { 120 } { 3 ! } x ^ { 3 } + \cdots , y ( 3 ) \approx 601.000$

Question

Quizplus · Accepted Answer

We are given the differential equation $y ^ {t } - 2 x y = 0$ and the initial condition $y ( 0 ) = 1$. To find the series solution, we can use Taylor's Theorem which states that any sufficiently smooth function can be approximated by a Taylor series expansion about some point. In this case, we expand the function $y(x)$ about $x=0$ as
$$ y ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { y ^ { ( n ) } ( 0 ) } { n ! } x ^ { n } $$
where the derivatives of $y$ evaluated at $x=0$ can be found by repeatedly differentiating the differential equation with respect to $x$ and setting $x=0$. Thus, we have,
$$ y ( 0 ) = 1 \Rightarrow y ( 0 ) = 1, y ^ { \prime } ( 0 ) = 2 , y ^ { \prime \prime } ( 0 ) = 4 , y ^ { \prime \prime \prime } ( 0 ) = 16 , \ldots $$
Now, we substitute these values into the Taylor series expression and simplify to obtain:
$$ y ( x ) = 1 + \frac { 2 } { 2 ! } x ^ { 2 } + \frac { 12 } { 4 ! } x ^ { 4 } + \frac { 120 } { 6 ! } x ^ { 6 } + \cdots $$
To find $y(3)$, we simply substitute $x=3$ into the series above and use the first four terms to obtain:
$$ y ( 3 ) \approx 1 + \frac { 2 } { 2 ! } ( 3 ) ^ { 2 } + \frac { 12 } { 4 ! } ( 3 ) ^ { 4 } + \frac { 120 } { 6 ! } ( 3 ) ^ { 6 } \approx 172.000 $$
Therefore, the best choice is option D.

Taylor's Theorem to Find the First Four Terms of the Series