Consider the following differential equation $tobe$ solved using a power series.

$y^{\text{'}\text{'}}+xy=0$

Using the substitution $y={\displaystyle \underset{n=0}{\overset{}{}}}{c}_n{x}^n,$ find $an$ expression for $c_{k+2}in$ terms $of{c}_{k-1}$ for $k=1,2,3$dots

$c_{k+2}=$

Find two power series solutions $of$ the given differential equation about the ordinary point $x=0.$

$y_1=1+x^2+\frac{x^3}{6}+$dots and $y_2=x+x^2+\frac{x^4}{12}+$dots

$y_1=1-\frac{x^2}{2}+\frac{x^4}{24}-$dots and $y_2=x-\frac{x^3}{6}+\frac{x^5}{120}-$dots

$y_1=1+\frac{x^2}{2}+\frac{x^4}{24}+$dots and $y_2=x+\frac{x^3}{6}+\frac{x^5}{120}+$dots

$y_1=1-\frac{x^3}{6}+\frac{x^6}{180}-$dots and $y_2=x-\frac{x^4}{12}+\frac{x^7}{504}-$dots

$y_1=1+\frac{x^3}{6}+\frac{x^6}{180}-$dots and $y_2=x+\frac{x^4}{12}+\frac{x^7}{504}-$dots