Elementary Differential Equations with Boundary Value Problems

Werner E. Kohler, Lee W. Johnson

Chapter 8

Series Solutions of Linear Differential Equations - all with Video Answers

Educators

Section 1

Introduction

Problem 1

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=0}^{\infty} \frac{t^{n}}{2^{n}}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 2

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=1}^{\infty} \frac{t^{n}}{n^{2}}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 3

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=0}^{\infty}(t-2)^{n}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 4

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=0}^{\infty}(3 t-1)^{n}
$$

William Semus

Numerade Educator

Problem 5

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=0}^{\infty} \frac{(t-1)^{n}}{n !}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 6

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=0}^{\infty} n !(t-1)^{n}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 7

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=1}^{\infty} \frac{(-1)^{n} t^{n}}{n}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 8

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=0}^{\infty} \frac{(-1)^{n}(t-3)^{n}}{4^{n}}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 9

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=1}^{\infty}(\ln n)(t+2)^{n}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 10

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=0}^{\infty} n^{3}(t-1)^{n}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 11

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=0}^{\infty} \frac{\sqrt{n}}{2^{n}}(t-4)^{n}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 12

As in Example 1, use the ratio test to find the radius of convergence $R$ for the given power series.
$$
\sum_{n=1}^{\infty} \frac{(t-2)^{n}}{\arctan n}
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 13

In each exercise, functions $f(t)$ and $g(t)$ are given. The functions $f(t)$ and $g(t)$ are defined by a power series that converges in $-R<t-t_{0}<R$, where $R$ is a positive constant. In each exercise, determine the largest value $R$ such that $f(t)$ and $g(t)$ both converge in $-R<t-t_{0}<R$. In addition,
(a) Write out the first four terms of the power series for $f(t)$ and $g(t)$.
(b) Write out the first four terms of the power series for $f(t)+g(t)$.
(c) Write out the first four terms of the power series for $f(t)-g(t)$.
(d) Write out the first four terms of the power series for $f^{\prime}(t)$.
(e) Write out the first four terms of the power series for $f^{\prime \prime}(t)$.
$$
f(t)=\sum_{n=0}^{\infty} t^{n}, \quad g(t)=\sum_{n=0}^{\infty} n^{2} t^{n}
$$

Uma Kumari

Numerade Educator

Problem 14

In each exercise, functions $f(t)$ and $g(t)$ are given. The functions $f(t)$ and $g(t)$ are defined by a power series that converges in $-R<t-t_{0}<R$, where $R$ is a positive constant. In each exercise, determine the largest value $R$ such that $f(t)$ and $g(t)$ both converge in $-R<t-t_{0}<R$. In addition,
(a) Write out the first four terms of the power series for $f(t)$ and $g(t)$.
(b) Write out the first four terms of the power series for $f(t)+g(t)$.
(c) Write out the first four terms of the power series for $f(t)-g(t)$.
(d) Write out the first four terms of the power series for $f^{\prime}(t)$.
(e) Write out the first four terms of the power series for $f^{\prime \prime}(t)$.
$$
f(t)=\sum_{n=0}^{\infty} n t^{n}, \quad g(t)=\sum_{n=0}^{\infty}(-1)^{n} n t^{n}
$$

Uma Kumari

Numerade Educator

Problem 15

In each exercise, functions $f(t)$ and $g(t)$ are given. The functions $f(t)$ and $g(t)$ are defined by a power series that converges in $-R<t-t_{0}<R$, where $R$ is a positive constant. In each exercise, determine the largest value $R$ such that $f(t)$ and $g(t)$ both converge in $-R<t-t_{0}<R$. In addition,
(a) Write out the first four terms of the power series for $f(t)$ and $g(t)$.
(b) Write out the first four terms of the power series for $f(t)+g(t)$.
(c) Write out the first four terms of the power series for $f(t)-g(t)$.
(d) Write out the first four terms of the power series for $f^{\prime}(t)$.
(e) Write out the first four terms of the power series for $f^{\prime \prime}(t)$.
$$
f(t)=\sum_{n=0}^{\infty}(-1)^{n} 2^{n}(t-1)^{n}, \quad g(t)=\sum_{n=0}^{\infty}(t-1)^{n}
$$

Uma Kumari

Numerade Educator

Problem 16

In each exercise, functions $f(t)$ and $g(t)$ are given. The functions $f(t)$ and $g(t)$ are defined by a power series that converges in $-R<t-t_{0}<R$, where $R$ is a positive constant. In each exercise, determine the largest value $R$ such that $f(t)$ and $g(t)$ both converge in $-R<t-t_{0}<R$. In addition,
(a) Write out the first four terms of the power series for $f(t)$ and $g(t)$.
(b) Write out the first four terms of the power series for $f(t)+g(t)$.
(c) Write out the first four terms of the power series for $f(t)-g(t)$.
(d) Write out the first four terms of the power series for $f^{\prime}(t)$.
(e) Write out the first four terms of the power series for $f^{\prime \prime}(t)$.
$$
f(t)=\sum_{n=0}^{\infty} 2^{n}(t+1)^{n}, \quad g(t)=\sum_{n=0}^{\infty} n(t+1)^{n}
$$

Uma Kumari

Numerade Educator

Problem 17

By shifting the index of summation as in equation $(9)$ or $(13)$, rewrite the given power series so that the general term involves $t^{n}$.
$$
\sum_{n=0}^{\infty} 2^{n} t^{n+2}
$$

Nick Johnson

Numerade Educator

Problem 18

By shifting the index of summation as in equation $(9)$ or $(13)$, rewrite the given power series so that the general term involves $t^{n}$.
$$
\sum_{n=0}^{\infty}(n+1)(n+2) t^{n+3}
$$

Nick Johnson

Numerade Educator

Problem 19

By shifting the index of summation as in equation $(9)$ or $(13)$, rewrite the given power series so that the general term involves $t^{n}$.
$$
\sum_{n=0}^{\infty} a_{n} t^{n+2}
$$

Nick Johnson

Numerade Educator

Problem 20

By shifting the index of summation as in equation $(9)$ or $(13)$, rewrite the given power series so that the general term involves $t^{n}$.
$$
\sum_{n=1}^{\infty} n a_{n} t^{n-1}
$$

Nick Johnson

Numerade Educator

Problem 21

By shifting the index of summation as in equation $(9)$ or $(13)$, rewrite the given power series so that the general term involves $t^{n}$.
$$
\sum_{n=2}^{\infty} n(n-1) a_{n} t^{n-2}
$$

Nick Johnson

Numerade Educator

Problem 22

By shifting the index of summation as in equation $(9)$ or $(13)$, rewrite the given power series so that the general term involves $t^{n}$.
$$
\sum_{n=0}^{\infty}(-1)^{n} a_{n} t^{n+3}
$$

Nick Johnson

Numerade Educator

Problem 23

By shifting the index of summation as in equation $(9)$ or $(13)$, rewrite the given power series so that the general term involves $t^{n}$.
$$
\sum_{n=0}^{\infty}(-1)^{n+1}(n+1) a_{n} t^{n+2}
$$

Nick Johnson

Numerade Educator

Problem 24

Using the information given in (7), write a Maclaurin series for the given function $f(t)$. Determine the radius of convergence of the series.
$$
f(t)=t^{2}(t-\sin t)
$$

Yaw Asomani

Numerade Educator

Problem 25

Using the information given in (7), write a Maclaurin series for the given function $f(t)$. Determine the radius of convergence of the series.
$$
f(t)=1-\cos (3 t)
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 26

Using the information given in (7), write a Maclaurin series for the given function $f(t)$. Determine the radius of convergence of the series.
$$
f(t)=\frac{1}{1+2 t}
$$

Yiming Zhang

Numerade Educator

Problem 27

Using the information given in (7), write a Maclaurin series for the given function $f(t)$. Determine the radius of convergence of the series.
$$
f(t)=\frac{1}{1-t^{2}}
$$

Yiming Zhang

Numerade Educator

Problem 28

Use series (7a) to determine the first four nonvanishing terms of the Maclaurin series for
(a) $\sinh t=\frac{e^{t}-e^{-t}}{2}$
(b) $\cosh t=\frac{e^{t}+e^{-t}}{2}$

Nick Johnson

Numerade Educator

Problem 29

Consider the differential equation $y^{\prime \prime}-\omega^{2} y=0$, where $\omega$ is a positive constant. As in Example 2, assume this differential equation has a solution of the form $y(t)=\sum_{n=0}^{\infty} a_{n} t^{n} .$
(a) Determine a recurrence relation for the coefficients $a_{0}, a_{1}, a_{2}, \ldots$
(b) As in equation (12), express the general solution in the form
$$
y(t)=a_{0} y_{1}(t)+\left(\frac{a_{1}}{\omega}\right) y_{2}(t) .
$$
What are the functions $y_{1}(t)$ and $y_{2}(t)$ ? [Hint: Recall the series in Exercise 28.]

Nadir Musofer

Numerade Educator

Problem 30

In each exercise,
(a) Use the given information to determine a power series representation of the function $y(t)$.
(b) Determine the radius of convergence of the series found in part (a).
(c) Where possible, use (7) to identify the function $y(t)$.
$$
y^{\prime}(t)=\sum_{n=1}^{\infty} n t^{n-1}=1+2 t+3 t^{2}+\cdots, \quad y(0)=1
$$

Adrian Co

Numerade Educator

Problem 31

In each exercise,
(a) Use the given information to determine a power series representation of the function $y(t)$.
(b) Determine the radius of convergence of the series found in part (a).
(c) Where possible, use (7) to identify the function $y(t)$.
$$
y^{\prime}(t)=\sum_{n=0}^{\infty} \frac{(t-1)^{n}}{n !}=1+(t-1)+\frac{(t-1)^{2}}{2 !}+\cdots, \quad y(1)=1
$$

Adrian Co

Numerade Educator

Problem 32

In each exercise,
(a) Use the given information to determine a power series representation of the function $y(t)$.
(b) Determine the radius of convergence of the series found in part (a).
(c) Where possible, use (7) to identify the function $y(t)$.
$$
y^{\prime \prime}(t)=\sum_{n=0}^{\infty}(-1)^{n} \frac{t^{n}}{n !}=1-t+\frac{t^{2}}{2 !}-\frac{t^{3}}{3 !}+\cdots, \quad y(0)=1, \quad y^{\prime}(0)=-1
$$

Adrian Co

Numerade Educator

Problem 33

In each exercise,
(a) Use the given information to determine a power series representation of the function $y(t)$.
(b) Determine the radius of convergence of the series found in part (a).
(c) Where possible, use (7) to identify the function $y(t)$.
$$
y^{\prime}(t)=\sum_{n=2}^{\infty}(-1)^{n} \frac{(t-1)^{n}}{n !}=\frac{(t-1)^{2}}{2 !}-\frac{(t-1)^{3}}{3 !}+\frac{(t-1)^{4}}{4 !}-\frac{(t-1)^{5}}{5 !}+\cdots, y(1)=0
$$

Adrian Co

Numerade Educator

Problem 34

In each exercise,
(a) Use the given information to determine a power series representation of the function $y(t)$.
(b) Determine the radius of convergence of the series found in part (a).
(c) Where possible, use (7) to identify the function $y(t)$.
$$
y(t)=\int_{0}^{t} f(s) d s, \text { where } f(s)=\sum_{n=0}^{\infty}(-1)^{n} s^{2 n}=1-s^{2}+s^{4}-s^{6}+\cdots
$$

Adrian Co

Numerade Educator

Problem 35

In each exercise,
(a) Use the given information to determine a power series representation of the function $y(t)$.
(b) Determine the radius of convergence of the series found in part (a).
(c) Where possible, use (7) to identify the function $y(t)$.
$$
\int_{0}^{t} y(s) d s=\sum_{n=1}^{\infty} \frac{t^{n}}{n}=t+\frac{t^{2}}{2}+\frac{t^{3}}{3}+\cdots
$$

Adrian Co

Numerade Educator

Problem 36

In each exercise, an initial value problem is given. Assume that the initial value problem has a solution of the form $y(t)=\sum_{n=0}^{\infty} a_{n} t^{n}$, where the series has a positive radius of convergence. Determine the first six coefficients, $a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$. Note that $y(0)=a_{0}$ and that $y^{\prime}(0)=a_{1}$. Thus, the initial conditions determine the arbitrary constants. In Exercises 40 and 41 , the exact solution is given in terms of exponential functions. Check your answer by comparing it with the Maclaurin series expansion of the exact solution.
$$
y^{\prime \prime}-t y^{\prime}-y=0, \quad y(0)=1, \quad y^{\prime}(0)=-1
$$

Lucas Stefanic

Numerade Educator

Problem 37

In each exercise, an initial value problem is given. Assume that the initial value problem has a solution of the form $y(t)=\sum_{n=0}^{\infty} a_{n} t^{n}$, where the series has a positive radius of convergence. Determine the first six coefficients, $a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$. Note that $y(0)=a_{0}$ and that $y^{\prime}(0)=a_{1}$. Thus, the initial conditions determine the arbitrary constants. In Exercises 40 and 41 , the exact solution is given in terms of exponential functions. Check your answer by comparing it with the Maclaurin series expansion of the exact solution.
$$
y^{\prime \prime}+t y^{\prime}-2 y=0, \quad y(0)=0, \quad y^{\prime}(0)=1
$$

Lucas Stefanic

Numerade Educator

Problem 38

In each exercise, an initial value problem is given. Assume that the initial value problem has a solution of the form $y(t)=\sum_{n=0}^{\infty} a_{n} t^{n}$, where the series has a positive radius of convergence. Determine the first six coefficients, $a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$. Note that $y(0)=a_{0}$ and that $y^{\prime}(0)=a_{1}$. Thus, the initial conditions determine the arbitrary constants. In Exercises 40 and 41 , the exact solution is given in terms of exponential functions. Check your answer by comparing it with the Maclaurin series expansion of the exact solution.
$$
y^{\prime \prime}+t y=0, \quad y(0)=1, \quad y^{\prime}(0)=2
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 39

In each exercise, an initial value problem is given. Assume that the initial value problem has a solution of the form $y(t)=\sum_{n=0}^{\infty} a_{n} t^{n}$, where the series has a positive radius of convergence. Determine the first six coefficients, $a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$. Note that $y(0)=a_{0}$ and that $y^{\prime}(0)=a_{1}$. Thus, the initial conditions determine the arbitrary constants. In Exercises 40 and 41 , the exact solution is given in terms of exponential functions. Check your answer by comparing it with the Maclaurin series expansion of the exact solution.
$$
y^{\prime \prime}+(1+t) y^{\prime}+y=0, \quad y(0)=-1, \quad y^{\prime}(0)=1
$$

Lucas Stefanic

Numerade Educator

Problem 40

In each exercise, an initial value problem is given. Assume that the initial value problem has a solution of the form $y(t)=\sum_{n=0}^{\infty} a_{n} t^{n}$, where the series has a positive radius of convergence. Determine the first six coefficients, $a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$. Note that $y(0)=a_{0}$ and that $y^{\prime}(0)=a_{1}$. Thus, the initial conditions determine the arbitrary constants. In Exercises 40 and 41 , the exact solution is given in terms of exponential functions. Check your answer by comparing it with the Maclaurin series expansion of the exact solution.
$$
y^{\prime \prime}-5 y^{\prime}+6 y=0, \quad y(0)=1, \quad y^{\prime}(0)=2, \quad y(t)=e^{2 t}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 41

In each exercise, an initial value problem is given. Assume that the initial value problem has a solution of the form $y(t)=\sum_{n=0}^{\infty} a_{n} t^{n}$, where the series has a positive radius of convergence. Determine the first six coefficients, $a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$. Note that $y(0)=a_{0}$ and that $y^{\prime}(0)=a_{1}$. Thus, the initial conditions determine the arbitrary constants. In Exercises 40 and 41 , the exact solution is given in terms of exponential functions. Check your answer by comparing it with the Maclaurin series expansion of the exact solution.
$$
y^{\prime \prime}-2 y^{\prime}+y=0, \quad y(0)=0, \quad y^{\prime}(0)=2, \quad y(t)=2 t e^{t}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator