Differential Equations and Linear Algebra

Stephen W. Goode, Scott A. Annin

Chapter 11

Series Solutions to Linear Differential Equations - all with Video Answers

Educators

Section 1

A Review of Power Series

Problem 1

Determine the radius of convergence of the given power series.
$$\sum_{n=0}^{\infty} \frac{x^{n}}{2^{2 n}}$$

Sandro Maludze

Sandro Maludze

Numerade Educator

Problem 2

Determine the radius of convergence of the given power series.
$$\sum_{n=0}^{\infty} \frac{(3 x)^{n}}{5^{3 n}}$$

Sandro Maludze

Numerade Educator

Problem 3

Determine the radius of convergence of the given power series.
$$\sum_{n=0}^{\infty} \frac{x^{n}}{n^{2}}$$

Sandro Maludze

Numerade Educator

Problem 4

Determine the radius of convergence of the given power series.
$$\sum_{n=0}^{\infty} \frac{2^{n} x^{n}}{n}$$

Sandro Maludze

Numerade Educator

Problem 5

Determine the radius of convergence of the given power series.
$$\sum_{n=0}^{\infty} n ! x^{n}$$

Sandro Maludze

Numerade Educator

Problem 6

Determine the radius of convergence of the given power series.
$$\sum_{n=0}^{\infty} \frac{5^{n} x^{n}}{n !}$$

Sandro Maludze

Numerade Educator

Problem 7

Determine the radius of convergence of the power series representation of the given function with center $x_{0}$.
$$f(x)=\frac{x^{2}-1}{x+2}, \quad x_{0}=0$$

Sandro Maludze

Numerade Educator

Problem 8

Determine the radius of convergence of the power series representation of the given function with center $x_{0}$.
$$f(x)=\frac{x}{x^{2}+1}, \quad x_{0}=0$$

Sandro Maludze

Numerade Educator

Problem 9

Determine the radius of convergence of the power series representation of the given function with center $x_{0}$.
$$f(x)=\frac{2 x}{x^{2}+16}, \quad x_{0}=1$$

Sandro Maludze

Numerade Educator

Problem 10

Determine the radius of convergence of the power series representation of the given function with center $x_{0}$.
$$f(x)=\frac{x^{2}-3}{x^{2}-2 x+5}, \quad x_{0}=0$$

Sandro Maludze

Numerade Educator

Problem 11

Determine the radius of convergence of the power series representation of the given function with center $x_{0}$.
$$f(x)=\frac{x}{\left(x^{2}+4 x+13\right)(x-3)}, \quad x_{0}=-1$$

Sandro Maludze

Numerade Educator

Problem 12

(a) Determine all values of $x$ at which the function
$f(x)=\frac{1}{x^{2}-1}$
is analytic.
(b) Determine the radius of convergence of a power series representation of the function $(11.1 .6)$ centered at $x=x_{0} .$ (You will need to consider the cases $-1<x_{0}<1$ and $\left|x_{0}\right|>1$ separately.

Adrian Co

Numerade Educator

Problem 13

By redefining the ranges of the summations appearing on the left-hand side, show that
$$
\begin{aligned}
\sum_{n=2}^{\infty} n(n-1) a_{n-1} x^{n-2} &+\sum_{n=1}^{\infty} n a_{n} x^{n-1} \\
&=\sum_{n=0}^{\infty}(n+1)(n+3) a_{n+1} x^{n}
\end{aligned}
$$

NM

Nicholas Ma

Numerade Educator

Problem 14

If $f(x)=\sum_{n=0}^{\infty} a_{n} x^{n},$ where the coefficients in the expansion satisfy
$$
\sum_{n=0}^{\infty} n(n+2) a_{n} x^{n}+\sum_{n=1}^{\infty}(n-3) a_{n-1} x^{n}=0
$$
determine $f(x)$

Thane Stiles

Numerade Educator

Problem 15

Suppose it is known that the coefficients in the expansion
$$
f(x)=\sum_{\Sigma=0}^{x} a_{x} x^{r}
$$
satisfy
$$
\sum_{n=0}^{\infty}(n+2) a_{n+1} x^{n}-\sum_{n=0}^{\infty} a_{n} x^{n}=0
$$
Show that
$$
f(x)=\frac{a_{0}}{x} \sum_{n=0}^{\infty} \frac{1}{(n+1) !} x^{n+1}
$$
and express this in terms of familiar elementary functions.

Thane Stiles

Numerade Educator

Problem 16

If
$$
\sum_{n=1}^{\infty}(n+1)(n+2) a_{n+1} x^{n}-\sum_{n=1}^{\infty} n a_{n-1} x^{n}=0
$$
show that for $k=1,2,3, \ldots,$ we have
$$
a_{2 k}=\frac{1 \cdot 3 \cdot 5 \cdots(2 k-1)}{(2 k+1) !} a_{0}, \quad a_{2 k+1}=\frac{2^{k+1} k !}{(2 k+2) !} a_{1}
$$.

NM

Nicholas Ma

Numerade Educator