Chapter 12.11, Problem 14P

Solve $y^{\prime \prime }=-y$ by the Frobenius method. You should find that the roots of the indicial equation are $s=0$ and $s=1.$ The value $s=0$ leads to the solutions $\cos x$ and $\sin x$ as you would expect. For $s=1,$ call the series $y={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{b}_n}{x}^{n+1},$ and find the relation $b_{n+2}=-\frac{b_n}{(n+3)(n+2)}$.

Show that the $b_0$ series obtained from this relation is just $\sin x,$ but that the $b_1$ series is not a solution of the differential equation. What is wrong?