• Home
  • Textbooks
  • Linear Algebra With Applications
  • Linear Differential Equations

Linear Algebra With Applications

Otto Bretscher

Chapter 9

Linear Differential Equations - all with Video Answers

Educators

Section 1

An Introduction to Continuous Dynamical Systems

01:49

Problem 1

Solve the initial value problems posed. Graph the solution.
$$\frac{d x}{d t}=5 x \text { with } x(0)=7$$

Diogo Caetano
Diogo Caetano
Numerade Educator
03:10

Problem 2

Solve the initial value problems posed. Graph the solution.
$$\frac{d x}{d t}=-0.71 x \text { with } x(0)=-e$$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
04:27

Problem 3

Solve the initial value problems posed. Graph the solution.
$$\frac{d P}{d t}=0.03 P \text { with } P(0)=7$$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
03:03

Problem 4

Solve the initial value problems posed. Graph the solution.
$$\frac{d y}{d t}=0.8 t \text { with } y(0)=-0.8$$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
02:57

Problem 5

Solve the initial value problems posed. Graph the solution.
$$\frac{d y}{d t}=0.8 y \text { with } y(0)=-0.8$$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
01:37

Problem 6

Solve the nonlinear differential equations using the method of separation of variables: Write the differential equation $d x / d t=f(x)$ as $d x / f(x)=d t$ and integrate both sides.
$$\frac{d x}{d t}=\frac{1}{x}, x(0)=1$$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
03:38

Problem 7

Solve the nonlinear differential equations using the method of separation of variables: Write the differential equation $d x / d t=f(x)$ as $d x / f(x)=d t$ and integrate both sides.
$\frac{d x}{d t}=x^{2}, x(0)=1 .$ Describe the behavior of your solution as $t$ increases.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
02:35

Problem 8

Solve the nonlinear differential equations using the method of separation of variables: Write the differential equation $d x / d t=f(x)$ as $d x / f(x)=d t$ and integrate both sides.
$$\frac{d x}{d t}=\sqrt{x}, x(0)=4$$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
01:37

Problem 9

Solve the nonlinear differential equations using the method of separation of variables: Write the differential equation $d x / d t=f(x)$ as $d x / f(x)=d t$ and integrate both sides.
$$\frac{d x}{d t}=x^{k}(\text { with } k \neq 1), x(0)=1$$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
01:38

Problem 10

Solve the nonlinear differential equations using the method of separation of variables: Write the differential equation $d x / d t=f(x)$ as $d x / f(x)=d t$ and integrate both sides.
$$\frac{d x}{d t}=\frac{1}{\cos (x)}, x(0)=0$$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
01:44

Problem 11

Solve the nonlinear differential equations using the method of separation of variables: Write the differential equation $d x / d t=f(x)$ as $d x / f(x)=d t$ and integrate both sides.
$$\frac{d x}{d t}=1+x^{2}, x(0)=0$$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
02:03

Problem 12

Find a differential equation of the form $d x / d t=k x$ for which $x(t)=3^{t}$ is a solution.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
07:25

Problem 13

In $1778,$ a wealthy Pennsylvanian merchant named Jacob DeHaven lent $\$ 450,000$ to the Continental Congress to support the troops at Valley Forge. The loan was never repaid. Mr. DeHaven's descendants have taken the U.S. government to court to collect what they believe they are owed. The going interest rate at the time was $6 \%$. How much were the DeHavens owed in 1990
a. if interest is compounded yearly?
b. if interest is compounded continuously

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
05:30

Problem 14

The carbon in living matter contains a minute proportion of the radioactive isotope $C-14 .$ This radiocarbon arises from cosmic-ray bombardment in the upper atmosphere and enters living systems by exchange processes. After the death of an organism, exchange stops, and the carbon decays. Therefore, carbon dating enables us to calculate the time at which an organism died. Let $x(t)$ be the proportion of the original $C-14$ still present $t$ years after death. By definition, $x(0)=1=100 \% .$ We are told that $x(t)$ satisfies the differential equation
\[
\frac{d x}{d t}=-\frac{1}{8270} x
\]
a. Find a formula for $x(t) .$ Determine the half-life of C-14 (that is, the time it takes for half of the $C-14$ to decay)
b. The Iceman. In 1991 , the body of a man was found in melting snow in the Alps of Northem Italy. A wellknown historian in Innsbruck, Austria, determined that the man had lived in the Bronze Age, which started about 2000 B.C. in that region. Examination of tissue samples performed independently at Zürich and Oxford revealed that $47 \%$ of the C-14 present in the body at the time of his death had decayed. When did this man die? Is the result of the carbon dating compatible with the estimate of the Austrian historian?

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
02:52

Problem 15

Justify the "Rule of $69 ":$ If a quantity grows at a constant instantaneous rate of $k \%$, then its doubling time is about $69 / k .$ Example: In 2008 the population of Madagascar was about 20 million, growing at an annual rate of about $3 \%,$ with a doubling time of about $69 / 3=23$ years.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
02:59

Problem 16

Consider the system
\[
\frac{d \vec{x}}{d t}=\left[\begin{array}{cc}
\lambda_{1} & 0 \\
0 & \lambda_{2}
\end{array}\right] \vec{x}
\]
For the values of $\lambda_{1}$ and $\lambda_{2}$ given, sketch the trajectories for all nine initial values shown in the following figure. For each of the points, trace out both the future and the past of the system.
$$\lambda_{1}=1, \lambda_{2}=-1$$

Nick Johnson
Nick Johnson
Numerade Educator
01:47

Problem 17

Consider the system
\[
\frac{d \vec{x}}{d t}=\left[\begin{array}{cc}
\lambda_{1} & 0 \\
0 & \lambda_{2}
\end{array}\right] \vec{x}
\]
For the values of $\lambda_{1}$ and $\lambda_{2}$ given, sketch the trajectories for all nine initial values shown in the following figure. For each of the points, trace out both the future and the past of the system.
$$\lambda_{1}=1, \lambda_{2}=2$$

Nick Johnson
Nick Johnson
Numerade Educator
01:47

Problem 18

Consider the system
\[
\frac{d \vec{x}}{d t}=\left[\begin{array}{cc}
\lambda_{1} & 0 \\
0 & \lambda_{2}
\end{array}\right] \vec{x}
\]
For the values of $\lambda_{1}$ and $\lambda_{2}$ given, sketch the trajectories for all nine initial values shown in the following figure. For each of the points, trace out both the future and the past of the system.
$$\lambda_{1}=-1, \lambda_{2}=-2$$

Nick Johnson
Nick Johnson
Numerade Educator
02:59

Problem 19

Consider the system
\[
\frac{d \vec{x}}{d t}=\left[\begin{array}{cc}
\lambda_{1} & 0 \\
0 & \lambda_{2}
\end{array}\right] \vec{x}
\]
For the values of $\lambda_{1}$ and $\lambda_{2}$ given, sketch the trajectories for all nine initial values shown in the following figure. For each of the points, trace out both the future and the past of the system.
$$\lambda_{1}=0, \lambda_{2}=1$$

Nick Johnson
Nick Johnson
Numerade Educator
01:47

Problem 20

Consider the system $d \vec{x} / d t=A \vec{x}$ with $A=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]$ Sketch a direction field for $A \bar{x}$. Based on your sketch, describe the trajectories geometrically. From your sketch. can you guess a formula for the solution with $\vec{x}_{0}=\left[\begin{array}{l}1 \\ 0\end{array}\right] ?$ Verify your guess by substituting into the equations.

Nick Johnson
Nick Johnson
Numerade Educator
02:14

Problem 21

Consider the system $d \vec{x} / d t=A \vec{x}$ with $A=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]$ Sketch a direction field of $A \vec{x} .$ Based on your sketch describe the trajectories geometrically. Can you find the solutions analytically?

Nick Johnson
Nick Johnson
Numerade Educator
01:32

Problem 22

Consider a linear system $d \vec{x} / d t=A \vec{x}$ of arbitrary size. Suppose $\vec{x}_{1}(t)$ and $\vec{x}_{2}(t)$ are solutions of the system. Is the sum $\vec{x}(t)=\vec{x}_{1}(t)+\vec{x}_{2}(t)$ a solution as well? How do you know?

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 23

Consider a linear system $d \vec{x} / d t=A \vec{x}$ of arbitrary size. Suppose $\vec{x}_{1}(t)$ is a solution of the system and $k$ is an arbitrary constant. Is $\vec{x}(t)=k \vec{x}_{1}(t)$ a solution as well? How do you know?

Nick Johnson
Nick Johnson
Numerade Educator
01:02

Problem 24

Let $A$ be an $n \times n$ matrix and $k$ a scalar. Consider the following two systems:
\[
\begin{array}{l}
\frac{d \vec{x}}{d t}=A \vec{x} \\
\frac{d \vec{c}}{d t}=\left(A+k I_{n}\right) \vec{c}
\end{array}
\]
Show that if $\vec{x}(t)$ is a solution of system (I), then $\vec{c}(t)=e^{k t} \vec{x}(t)$ is a solution of system (II).

Nick Johnson
Nick Johnson
Numerade Educator
02:11

Problem 25

Let $A$ be an $n \times n$ matrix and $k$ a scalar. Consider the following two systems:
\[
\begin{array}{l}
\frac{d \vec{x}}{d t}=A \vec{x} \\
\frac{d \vec{c}}{d t}=k A \vec{c}
\end{array}
\]
Show that if $\vec{x}(t)$ is a solution of system (I), then $\vec{c}(t)=\vec{x}(k t)$ is a solution of system (II). Compare the vector fields of the two systems.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 26

Solve the system with the given initial value.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{ll}
1 & 2 \\
3 & 0
\end{array}\right] \vec{x} \text { with } \vec{x}(0)=\left[\begin{array}{l}
7 \\
2
\end{array}\right]$$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 27

Solve the system with the given initial value.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{rr}
-4 & 3 \\
2 & -3
\end{array}\right] \vec{x} \text { with } \vec{x}(0)=\left[\begin{array}{l}
1 \\
0
\end{array}\right]$$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 28

Solve the system with the given initial value.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{ll}
4 & 3 \\
4 & 8
\end{array}\right] \vec{x} \text { with } \vec{x}(0)=\left[\begin{array}{l}
1 \\
1
\end{array}\right]$$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 29

Solve the system with the given initial value.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{ll}
1 & 2 \\
2 & 4
\end{array}\right] \vec{x} \text { with } \vec{x}(0)=\left[\begin{array}{l}
5 \\
0
\end{array}\right]$$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 30

Solve the system with the given initial value.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{ll}
1 & 2 \\
2 & 4
\end{array}\right] \vec{x} \text { with } \vec{x}(0)=\left[\begin{array}{r}
2 \\
-1
\end{array}\right]$$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 31

Solve the system with the given initial value.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{lll}
2 & 1 & 1 \\
1 & 3 & 3 \\
3 & 2 & 2
\end{array}\right] \vec{x} \text { with } \bar{x}(0)=\left[\begin{array}{r}
1 \\
-2 \\
1
\end{array}\right]$$

Victor Salazar
Victor Salazar
Numerade Educator
01:54

Problem 32

Sketch rough phase portraits for the dynamical systems given.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{ll}
1 & 2 \\
3 & 0
\end{array}\right] \vec{x}$$

Nick Johnson
Nick Johnson
Numerade Educator
02:28

Problem 33

Sketch rough phase portraits for the dynamical systems given.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{rr}
-4 & 3 \\
2 & -3
\end{array}\right]$$

Nick Johnson
Nick Johnson
Numerade Educator
02:02

Problem 34

Sketch rough phase portraits for the dynamical systems given.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{ll}
4 & 3 \\
4 & 8
\end{array}\right] \vec{x}$$

Nick Johnson
Nick Johnson
Numerade Educator
02:19

Problem 35

Sketch rough phase portraits for the dynamical systems given.
$$\frac{d \vec{x}}{d t}=\left[\begin{array}{ll}
1 & 2 \\
2 & 4
\end{array}\right]$$

Nick Johnson
Nick Johnson
Numerade Educator
02:37

Problem 36

Sketch rough phase portraits for the dynamical systems given.
$$\vec{x}(t+1)=\left[\begin{array}{ll}
0.9 & 0.2 \\
0.2 & 1.2
\end{array}\right] \vec{x}(t)$$

Nick Johnson
Nick Johnson
Numerade Educator
01:48

Problem 37

Sketch rough phase portraits for the dynamical systems given.
$$\vec{x}(t+1)=\left[\begin{array}{rr}
1 & 0.3 \\
-0.2 & 1.7
\end{array}\right] \vec{x}(t)$$

Nick Johnson
Nick Johnson
Numerade Educator
02:04

Problem 38

Sketch rough phase portraits for the dynamical systems given.
$$\vec{x}(t+1)=\left[\begin{array}{rr}
1.1 & 0.2 \\
-0.4 & 0.5
\end{array}\right] \vec{x}(t)$$

Nick Johnson
Nick Johnson
Numerade Educator
03:40

Problem 39

Sketch rough phase portraits for the dynamical systems given.
$$\vec{x}(t+1)=\left[\begin{array}{rr}
0.8 & -0.4 \\
0.3 & 1.6
\end{array}\right] \vec{x}(t)$$

Nick Johnson
Nick Johnson
Numerade Educator
09:34

Problem 40

Find a $2 \times 2$ matrix $A$ such that the system $d \vec{x} / d t=A \vec{x}$ has
\[
\vec{x}(t)=\left[\begin{array}{l}
2 e^{2 t}+3 e^{3 t} \\
3 e^{2 t}+4 e^{3 t}
\end{array}\right]
\]
as one of its solutions.

Ryan Williams
Ryan Williams
Numerade Educator
03:38

Problem 41

Consider a noninvertible $2 \times 2$ matrix $A$ with two distinct eigenvalues. (Note that one of the eigenvalues must be 0.) Choose two eigenvectors $\vec{v}_{1}$ and $\vec{v}_{2}$ with eigenvalues $\lambda_{1}=0$ and $\lambda_{2}$ as shown in the accompanying figure. Suppose $\lambda_{2}$ is negative. Sketch a phase portrait for the system $d \vec{x} / d t=A \vec{x},$ clearly indicating the shape and long-term behavior of the trajectories.

Nick Johnson
Nick Johnson
Numerade Educator
04:50

Problem 42

Consider the interaction of two species of animals in a habitat. We are told that the change of the populations $x(t)$ and $y(t)$ can be modeled by the equations
\[
\left|\begin{array}{l}
\frac{d x}{d t}=1.4 x-1.2 y \\
\frac{d y}{d t}=0.8 x-1.4 y
\end{array}\right|
\]
where time $t$ is measured in years.
a. What kind of interaction do we observe (symbiosis, competition, or predator-prey)?
b. Sketch a phase portrait for this system. From the nature of the problem, we are interested only in the first quadrant.
c. What will happen in the long term? Does the outcome depend on the initial populations? If so, how?

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 43

Answer the questions posed in Exercise 42 for the following system:
\[
\begin{array}{l}
\frac{d x}{d t}=5 x-y \\
\frac{d y}{d t}=-2 x+4 y
\end{array}
\]

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 44

Answer the questions posed in Exercise 42 for the fol lowing system:
\[
\left|\begin{array}{l}
\frac{d x}{d t}=x+4 y \\
\frac{d y}{d t}=2 x-y
\end{array}\right|
\]

Victor Salazar
Victor Salazar
Numerade Educator
03:29

Problem 45

Two herds of vicious animals are fighting each other to the death. During the fight, the populations $x(t)$ and $y(t)$ of the two species can be modeled by the following system: 2
\[
\begin{array}{l}
\frac{d x}{d t}=-4 y | \\
\frac{d y}{d t}=-x
\end{array}
\]
a. What is the significance of the constants -4 and -1 in these equations? Which species has the more vicious (or more efficient) fighters?
b. Sketch a phase portrait for this system.
c. Who wins the fight (in the sense that some individuals of that species are left while the other herd is eradicated)? How does your answer depend on the initial populations?

Nick Johnson
Nick Johnson
Numerade Educator
03:02

Problem 46

Repeat Exercise 45 for the system
\[
\begin{array}{l}
\frac{d x}{d t}=\quad-p y \\
\frac{d y}{d t}=-q x
\end{array}
\]
where $p$ and $q$ are two positive constants.

Nick Johnson
Nick Johnson
Numerade Educator
03:57

Problem 47

The interaction of two populations of animals is modeled by the differential equations
\[
\begin{array}{l}
\left|\frac{d x}{d t}=-x+k y\right| \\
\frac{d y}{d t}=k x-4 y |
\end{array}
\]
for some positive constant $k$
a. What kind of interaction do we observe? What is the practical significance of the constant $k ?$
b. Find the eigenvalues of the coefficient matrix of the system. What can you say about the signs of these eigenvalues? How does your answer depend on the value of the constant $k ?$
c. For each case you discussed in part (b), sketch rough phase portrait. What does each phase portrait tell you about the future of the two populations?

Nick Johnson
Nick Johnson
Numerade Educator
03:04

Problem 48

Repeat Exercise 47 for the system
\[
\begin{array}{l}
\frac{d x}{d t}=-x+k y \\
\frac{d y}{d t}=x-4 y
\end{array} |
\]
where $k$ is a positive constant.

Nick Johnson
Nick Johnson
Numerade Educator
05:18

Problem 49

Here is a continuous model of a person's glucose regulatory system. (Compare this with Exercise $7.1 .52 .$ ) Let $g(t)$ and $h(t)$ be the excess glucose and insulin concentrations in a person's blood. We are told that
\[
\left|\begin{array}{l}
\frac{d g}{d t}=-g-0.2 h \\
\frac{d h}{d t}=0.6 g-0.2 h
\end{array}\right|
\]
where time $t$ is measured in hours. After a heavy holiday dinner, we measure $g(0)=30$ and $h(0)=0 .$ Find closed formulas for $g(t)$ and $h(t) .$ Sketch the trajectory.

Sanjoy Chatterjee
Sanjoy Chatterjee
Numerade Educator
View

Problem 50

Consider a linear system $d \vec{x} / d t=A \vec{x},$ where $A$ is a $2 \times 2$ matrix that is diagonalizable over $\mathbb{R}$. When is the zero state a stable equilibrium solution? Give your answer in terms of the determinant and the trace of $A$

Victor Salazar
Victor Salazar
Numerade Educator
01:37

Problem 51

Let $\vec{x}(t)$ be a differentiable curve in $\mathbb{R}^{n}$ and $S$ an $n \times n$ matrix. Show that
\[
\frac{d}{d t}(S \vec{x})=S \frac{d \vec{x}}{d t}
\]

Nick Johnson
Nick Johnson
Numerade Educator
02:26

Problem 52

Find all solutions of the system
\[
\frac{d \vec{x}}{d t}=\left[\begin{array}{ll}
\lambda & 1 \\
0 & \lambda
\end{array}\right] \vec{x}
\]
where $\lambda$ is an arbitrary constant. Hint: Exercises 21 and 24 are helpful. Sketch a phase portrait. For which choices of $\lambda$ is the zero state a stable equilibrium solution?

Nick Johnson
Nick Johnson
Numerade Educator
01:13

Problem 53

Solve the initial value problem
\[
\frac{d \vec{x}}{d t}=\left[\begin{array}{rr}
p & -q \\
q & p
\end{array}\right] \vec{x} \quad \text { with } \quad \vec{x}_{0}=\left[\begin{array}{l}
1 \\
0
\end{array}\right]
\]
Sketch the trajectory for the cases when $p$ is positive, negative, or $0 .$ In which cases does the trajectory approach the origin? (Hint: Exercises $20,24,$ and 25 are helpful.)

Steven Clarke
Steven Clarke
Numerade Educator
01:47

Problem 54

Consider a door that opens to only one side (as most doors do $) .$ A spring mechanism closes the door automatically. The state of the door at a given time $t$ (measured in seconds) is determined by the angular displacement $\theta(t)$ (measured in radians) and the angular velocity $\omega(t)=d \theta / d t .$ Note that $\theta$ is always positive or zero (since the door opens to only one side), but $\omega$ can be positive or negative (depending on whether the door is opening or closing.
When the door is moving freely (nobody is pushing or pulling), its movement is subject to the following differential equations:
$$\begin{array}{|l|l}
\frac{d \theta}{d t}=\quad \quad\quad\quad \omega & \text { (the definition of } \omega) \\
\frac{d \omega}{d t}=-2 \theta-3 \omega &(-2 \theta\text { reflects the force of the } \\& \text { spring, and }-3 \omega \text { models friction) }\end{array}$$
a. Sketch a phase portrait for this system.
b. Discuss the movement of the door represented by the qualitatively different trajectories. For which initial states does the door slam (i.e., reach $\theta=0$ with velocity $\omega<0$ )?

Manish Jain
Manish Jain
Numerade Educator
01:12

Problem 55

Answer the questions posed in Exercise 54 for the system
\[
\begin{array}{l}
\frac{d \theta}{d t}= \\
\frac{d \omega}{d t}=-p \theta-q \omega
\end{array}
\]
where $p$ and $q$ are positive, and $q^{2}>4 p$

Nick Johnson
Nick Johnson
Numerade Educator