Chapter 6.1, Problem 34E

In Problems 31–34 verify by direct substitution that the given power series is a solution of the indicated differential equation. [Hint: For a power $x^{2n+1}\text{let}k=n+1$.]

34. $y={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^n}{2^{2n}{\left(1!\right)}^2}{x}^{2n}},x{y}^{″}+y^{\prime }+xy=0$