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Fundamentals of Differential Equations

R. Kent Nagle

Chapter 8

Series Solutions of Differential Equations - all with Video Answers

Educators

Section 1

Introduction: The Taylor Polynomial Approximation

03:51

Problem 1

$$y^{\prime}=x^{2}+y^{2} ; \quad y(0)=1$$

Jack Chen
Jack Chen
Numerade Educator
03:05

Problem 2

$$y^{\prime}=y^{2} ; \quad y(0)=2$$

Roman Frolov
Roman Frolov
Numerade Educator
09:40

Problem 3

$$y^{\prime}=\sin y+e^{x} ; \quad y(0)=0$$

Roman Frolov
Roman Frolov
Numerade Educator
09:40

Problem 4

$$y^{\prime}=\sin (x+y) ; \quad y(0)=0$$

Roman Frolov
Roman Frolov
Numerade Educator
01:44

Problem 5

$$x^{\prime \prime}+t x=0 ; \quad x(0)=1, \quad x^{\prime}(0)=0$$

Jack Chen
Jack Chen
Numerade Educator
02:55

Problem 6

$$y^{\prime \prime}+y=0 ; \quad y(0)=0, \quad y^{\prime}(0)=1$$

Roman Frolov
Roman Frolov
Numerade Educator
02:29

Problem 7

$$\begin{array}{l}{y^{\prime \prime}(\theta)+y(\theta)^{3}=\sin \theta} \\ {y(0)=0, \quad y^{\prime}(0)=0}\end{array}$$

Jack Chen
Jack Chen
Numerade Educator
03:36

Problem 8

$$y^{\prime \prime}+\sin y=0 ; \quad y(0)=1, \quad y^{\prime}(0)=0$$

Roman Frolov
Roman Frolov
Numerade Educator
12:30

Problem 9

(a) Construct the Taylor polynomial $p_{3}(x)$ of degree 3 for the function $f(x)=\ln$ $x$ around $x=1$.
(b) Using the error formula (6), show that
$$
\left|\ln (1.5)-p_{3}(1.5)\right| \leq \frac{(0.5)^{4}}{4}=0.015625
$$
(c) Compare the estimate in part (b) with the actual error by calculating $$
\left|\ln (1.5)-p_{3}(1.5)\right|
$$
(d) Sketch the graphs of $\ln x$ and $p_{3}(x)$ (on the same axes) for 0<$x$<2.

Roman Frolov
Roman Frolov
Numerade Educator
12:30

Problem 10

(a) Construct the Taylor polynomial $p_{3}(x)$ of degree 3 for the function $f(x)=1 /(2-x)$ around $x=0$
$$$$(b) Using the error formula (6), show that
$$
\left|f\left(\frac{1}{2}\right)-p_{3}\left(\frac{1}{2}\right)\right|=\left|\frac{2}{3}-p_{3}\left(\frac{1}{2}\right)\right| \leq \frac{2}{3^{5}}$$
$$$$(c) Compare the estimate in part (b) with the actual error
$$
\left|\frac{2}{3}-p_{3}\left(\frac{1}{2}\right)\right|
$$
$$$$(d) Sketch the graphs of $1 /(2-x)$ and $p_{3}(x)$ (on the same axes) for $-2$<$x$<$2$

Roman Frolov
Roman Frolov
Numerade Educator
08:40

Problem 11

Argue that if $y=\phi(x)$ is a solution to the differential equation $y^{\prime \prime}+p(x) y^{\prime}+q(x) y=g(x)$ on the interval $(a, b)$, where $p, q$ and $g$ are each twice-differentiable, then the fourth derivative of $\phi(x)$ exists on $(a, b)$.

Roman Frolov
Roman Frolov
Numerade Educator
08:40

Problem 12

Argue that if $y=\phi(x)$ is a solution to the differential equation $y^{\prime \prime}+p(x) y^{\prime}+q(x) y=g(x)$ on the interval $(a, b)$, where $p, q$ and $g$ possess derivatives of all orders, then $\phi$ has derivatives of all orders on $(a, b)$.

Roman Frolov
Roman Frolov
Numerade Educator
06:57

Problem 13

Duffing's Equation. In the study of a nonlinear spring with periodic forcing, the following equation arises:
$$
y^{\prime \prime}+k y+r y^{3}=A \cos \omega t
$$
Let $k=r=A=1$ and $\omega=10$. Find the first three nonzero terms in the Taylor polynomial approximations to the solution with initial values $y(0)=0, y^{\prime}(0)=1$.

Carl David Cepeda
Carl David Cepeda
Numerade Educator
07:09

Problem 14

Soft versus Hard Springs. For Duffing's equation given in Problem 13, the behavior of the solutions changes as $r$ changes sign. When $r>0$, the restoring force $k y+r y^{3}$ becomes stronger than for the linear spring $(r=0)$. Such a spring is called hard. When $r<0$, the restoring force becomes weaker than the linear spring and the spring is called soft. Pendulums act like soft springs.
$$$$
(a) Redo Problem 13 with $r=-1$. Notice that for the initial conditions $y(0)=0, y^{\prime}(0)=1$, the soft and hard springs appear to respond in the same way for $t$ small.
$$$$
(b) Keeping $k=A=1$ and $\omega=10$, change the initial conditions to $y(0)=1$ and $y^{\prime}(0)=0$. Now redo Problem 13 with $r=\pm 1$.
$$$$
(c) Based on the results of part (b), is there a difference between the behavior of soft and hard springs for $t$ small? Describe.

Roman Frolov
Roman Frolov
Numerade Educator
03:44

Problem 15

The solution to the initial value problem
$$\begin{array}{l}{x y^{\prime \prime}(x)+2 y^{\prime}(x)+x y(x)=0} \\ {y(0)=1, \quad y^{\prime}(0)=0}\end{array}$$
has derivatives of all orders at $x=0$ (although this is far from obvious). Use L' Hopital's rule to compute the Taylor polynomial of degree 2 approximating this solution.

ST
Sydney Thomas
Numerade Educator
05:17

Problem 16

van der Pol Equation. In the study of the vacuum tube, the following equation is encountered:$$y^{\prime \prime}+(0.1)\left(y^{2}-1\right) y^{\prime}+y=0$$
Find the Taylor polynomial of degree 4 approximating the solution with the initial values $y(0)=1$, $y^{\prime}(0)=0$.

Roman Frolov
Roman Frolov
Numerade Educator