Chapter 5.1, Problem 15P

a.

To determine

The expression for $y^{\prime }$ and $y^{″}$ of $y={\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ with the first four terms of each series and also the coefficient of $x^n$ in the general term.

b.

To determine

To show: If $y^{″}=y$, then there are arbitrary coefficients $a_0$ and $a_1$ and hence obtain the terms $a_2$ and $a_3$ in terms of the arbitrary coefficients.

c.

To determine

To show: The expression $a_{n+2}=\frac{a_n}{\left(n+2\right)\left(n+1\right)},n=0,1,2,3,\dots$.