$H,1$

$,{?}_{pts}^{25}$

Consider the IVP $y^{\text{'}\text{'}}-xy=0$ where $y(0)=1$ and $y^{\text{'}}(0)=0.$ Using the power series centered $at0$

$y(x)={\displaystyle \underset{n=0}{\overset{}{}}}{a}_n{x}^n=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4+a_5{x}^5+a_6{x}^6+$cdots

determine

the $n^{th}$ pattern, $a_{n}n$ and one $or$ more previous $terms{x}^n$ for both series, and then note that one series starts $at$

$n=0$ and the other $n=1.$ Remember ${\displaystyle \underset{n=0}{\overset{}{}}}{b}_n{x}^n=b_0+{\displaystyle \underset{n=1}{\overset{}{}}}{b}_n{x}^n.$

the coefficients $a_0,a_1,a_2,a_3,a_4,a_5,$ and $a_6.$