Introductory Differential Equations

Martha L. Abell, James P. Braselton

Chapter 3 Applications Of First-order Differential Equations - all with Video Answers

Educators

Section 1

Population Growth And Decay

01:28

Problem 1

Suppose that a culture of bacteria has an initial population of $n=100$. If the population doubles every 3 days, determine the number of bacteria present after 30 days. How much time is required for the population to reach 4250 in number?

Edwin Bakalemwa

Numerade Educator

01:49

Problem 2

Suppose that the population in a yeast culture triples every 7 days. What is the population after 35 days? How much time is required for the population to be 10 times the initial population?

Amy Jiang

Numerade Educator

01:45

Problem 3

Suppose that two-thirds of the cells in a culture remain after 1 day. Use this information to determine the number of days until only one-third of the initial population remains.

Nick Johnson

Numerade Educator

04:07

Problem 4

Consider a radioactive substance with half-life 10 days. If there are initially $5000 \mathrm{~g}$ of the substance, how much remains after 365 days?

Sharfa Farzandh

Numerade Educator

01:20

Problem 5

Suppose that the half-life of an element is $1000 \mathrm{~h}$. If there are initially $100 \mathrm{~g}$, how much remains after $1 \mathrm{~h}$ ? How much remains after $500 \mathrm{~h}$ ?

Narayan Hari

Numerade Educator

06:52

Problem 6

Suppose that the population of a small town is initially 5000 . Due to the construction of an interstate highway, the population doubles over the next year. If the rate of growth is proportional to the current population, when will the population reach 25,000 ? What is the population after 5 years?

Willis James

Numerade Educator

02:05

Problem 7

Suppose that mold grows at a rate proportional to the amount present. If there are initially $500 \mathrm{~g}$ of mold and $6 \mathrm{~h}$ later there are $600 \mathrm{~g}$, determine the amount of mold present after 1 day. When is the amount of mold $1000 \mathrm{~g}$ ?

Prashansha Kaushik

Numerade Educator

05:27

Problem 8

Suppose that the rabbit population on a small island grows at a rate proportional to the number of rabbits present. If this population doubles after 100 days, when does the population triple?

Dj Tan

Numerade Educator

07:12

Problem 9

In a chemical reaction, chemical $\mathrm{A}$ is converted to chemical $B$ at a rate proportional to the amount of chemical A present. If half of chemical A remains after $5 \mathrm{~h}$, when does $1 / 6$ of the initial amount of chemical A remain? How much of the initial amount remains after $15 \mathrm{~h}$ ?

Dj Tan

Numerade Educator

02:33

Problem 10

If $90 \%$ of the initial amount of a radioactive element remains after 1 day, what is the half-life of the element?

Mike Gaerlan

Numerade Educator

01:33

Problem 11

If $y(t)$ represents the percent of a radioactive element that is present at time $t$ and the values of $y\left(t_{1}\right)$ and $y\left(t_{2}\right)$ are known, show that the half-life $H$ is given by
$$
H=\frac{\left(t_{2}-t_{1}\right) \ln 2}{\ln \left(y\left(t_{1}\right)\right)-\ln \left(y\left(t_{2}\right)\right)} .
$$

Sid Wan

University of Louisville

02:06

Problem 12

The half-life of ${ }^{14} \mathrm{C}$ is 5730 years. If the original amount of ${ }^{14} \mathrm{C}$ in a particular living organism is $20 \mathrm{~g}$ and that found in a fossil of that organism is $0.01 \mathrm{~g}$, determine the approximate age of the fossil.

Sriram Soundarrajan

Numerade Educator

01:06

Problem 13

After 10 days, $800 \mathrm{~g}$ of a radioactive element remain, and after 15 days, $560 \mathrm{~g}$ remain. What is the half-life of this element?

Joseph Liao

Numerade Educator

04:07

Problem 14

After 1 week, $10 \%$ of the initial amount of a radioactive element decays. How much decays after 2 weeks? When does half of the original amount decay?

Sharfa Farzandh

Numerade Educator

01:40

Problem 15

Determine the percentage of the original amount of ${ }^{226} \mathrm{Ra}$ that remains after 100 years.

Ankit Gupta

Numerade Educator

01:16

Problem 16

If an artifact contains $40 \%$ of the amount of 230 Th as a present-day sample, what is the age of the artifact?

Jorge Villanueva

Numerade Educator

01:32

Problem 17

On an archeological dig, scientists find an ancient tool near a fossilized human bone. If the tool and fossil contain $65 \%$ and $60 \%$ of the amount of ${ }^{14} \mathrm{C}$ as that in present-day samples, respectively, determine if the tool could have been used by the human.

Heather Zimmers

Numerade Educator

00:39

Problem 18

A certain group of people with initial population 10,000 grows at a rate proportional to the number present. The population doubles in 5 years. In how many years will the population triple?

Joseph Liao

Numerade Educator

05:15

Problem 19

Solve the logistic equation, $\mathrm{d} y / \mathrm{d} t=\alpha y(1-(1 / K) y)$, by viewing it as a Bernoulli equation and then solve the resulting linear equation by using an integrating factor rather than the method of undetermined coefficients that is illustrated in the examples.

Mohamed Raafat Mohamed

Numerade Educator

01:17

Problem 20

What is the limiting population, $\lim _{t \rightarrow \infty} y(t)$, of the United States population using the result obtained in Example 3.1.5?

Chris Bolognese

Numerade Educator

14:50

Problem 21

Solve the logistic equation if $r=1 / 100$ and $a=10^{-8}$ given that $y(0)=100,000$. Find $y(25)$. What is the limiting population?

Yaw Asomani

Numerade Educator

09:19

Problem 22

Five college students with the flu virus return to an isolated campus of 2500 students. If the rate at which this virus spreads is proportional to the number of infected students $y$ and to the number not infected $2500-y$, solve the IVP $\mathrm{d} y / \mathrm{d} t=k y(2500-y), y(0)=5$ to find the number of infected students after $t$ days if 25 students have the virus after 1 day. How many students have the flu after 5 days?

Mohamed Raafat Mohamed

Numerade Educator

08:38

Problem 23

One student in a college organization of 200 members proceeds to spread a rumor. If the rate at which this rumor spreads is proportional to the number of students $y$ that know about the rumor as well as the number that do not know, then solve the IVP to find the number of students informed of the rumor after $t$ days if 50 students are informed after 1 day. How many students know the rumor after 2 days? Will all of the students eventually be informed of the rumor? (see Exercise 22).

Mohamed Raafat Mohamed

Numerade Educator

05:18

Problem 24

Suppose that glucose enters the bloodstream at the constant rate of $r$ grams per minute while it is removed at a rate proportional to the amount $y$ present at any time. Solve the IVP dy/d $t=r-k y, y(0)=y_{0}$ to find $y(t)$. What is the eventual concentration of glucose in the bloodstream according to this model?

Sanjoy Chatterjee

Numerade Educator

01:48

Problem 25

What is the concentration of glucose in the bloodstream after $10 \mathrm{~min}$ if $r=5 \mathrm{~g} / \mathrm{min}$, $k=5$, and the initial concentration is $y(0)=500$ ? After 20 min? Does the concentration appear to reach its limiting value quickly or slowly? (see Exercise 24).

James Kiss

Numerade Educator

04:12

Problem 26

Suppose that we deposit a sum of money in a money market fund that pays interest at an annual rate $k$, and let $S(t)$ represent the value of the investment at time $t$. If the compounding takes place continuously, then the rate at which the value of the investment changes is the interest rate times the value of the investment, $\mathrm{d} S / \mathrm{d} t=k S$.
Use this equation to find $S(t)$ if $S(0)=S_{0}$.

Harshita Goel

Numerade Educator

02:19

Problem 27

Banks use different methods to compound interest. If the interest rate is $k$, and if interest is compounded $m$ times per year, then $S(t)=S_{0}(1+k / m)^{m t}$. When interest is compounded continuously, then $m \rightarrow \infty$. Compare
$$
\lim _{m \rightarrow \infty} S_{0}(1+k / m)^{m t}
$$
to the formula obtained in Exercise 26 .

Timothy Morris

Numerade Educator

09:16

Problem 28

(Dating Works of Art) We can determine if a work of art is more than 100 years old by determining if the lead-bearing materials
1B. Keisch, Dating works of art through their natural radioactivity: improvements and applications, Science, 160, 1968, 413-415.
contained in the work were manufactured within the last 100 years. The half-life of lead-210 $\left({ }^{210} \mathrm{~Pb}\right)$ is 22 years, while the half-life of radium-226 ( $\left.{ }^{226} \mathrm{Ra}\right)$ is 1700 years. Let SF denote the ratio of ${ }^{210} \mathrm{~Pb}$ to ${ }^{226}$ Ra per unit mass of lead. The approximate value of SF for works of art created in the last 80 years is 100 . Then, the quantity of lead $(1-\mathrm{Ra}) /(\mathrm{Po})$ at time $t$ is given by
$$
\frac{1-\mathrm{Ra}}{\text { Po }}=\frac{(\mathrm{SF}-1) \mathrm{e}^{-\lambda t}}{1+(\mathrm{SF}-1) \mathrm{e}^{-\lambda t}},
$$
where $\lambda$ is the disintegration constant for ${ }^{210} \mathrm{~Pb}^{1}{ }^{1}$ On the other hand, for very old paintings $(1-\mathrm{Ra}) /(\mathrm{Po}) \approx 0$.
(a) Determine the disintegration constant $\lambda$ for ${ }^{210} \mathrm{~Pb}$ where the amount of ${ }^{210} \mathrm{~Pb}$ at time $t$ is $y=y_{0} \mathrm{e}^{-\lambda t}$.
(b) Graph (1-Ra)/(Po) for $0 \leq t \leq 250$ using $\mathrm{SF}=100$.
The following table shows the ratio of ( $1-\mathrm{Ra}) /(\mathrm{Po}$ ) for various famous paintings.
The last two paintings, Lace Maker and Laughing Girl, were painted by the Dutch painter Jan Vermeer, who lived from 1632 to $1675 .$
(c) Determine if it is likely that any of the first six paintings were also painted by Vermeer (which would make them very valuable!). If not, approximate when they were painted.
The Gompertz equation is named after the English mathematician and statistician Benjamin Gompertz (1779-1865). He is most famous for Gompertz's Law of Mortality that was published in $1825 .$

Ben Nicholson

Numerade Educator

13:33

Problem 29

Consider the differential equation $\mathrm{d} y / \mathrm{d} t=-r(1-y / A) y$, where $r$ and $A$ are positive constants. (a) Find the equilibrium solutions, sketch the phase line, and classify the equilibrium solutions. (b) Describe how this equation differs from the logistic equation. (c) If $A=2$ and $y(0)=1$, what is $\lim _{t \rightarrow \infty} y(t)$ ? (d) If $A=2$ and $y(0)=3$, what is $\lim _{t \rightarrow \infty} y(t)$ ?

Melvin Adkins

Numerade Educator

03:20

Problem 30

(Gompertz's Law of Mortality) (a) Find and classify the equilibrium solutions of the Gompertz equation, $\mathrm{d} y / \mathrm{d} t=y(r-a \ln y)$, where $r$ and $a$ are positive constants.
(b) Gompertz's Law of Mortality states that $N^{\prime}(t)=r N(t) \ln (K / N(t))$, where $N(t)$ is the size of the population at time $t, r$ the growth rate, and $K$ the equilibrium population size. Show that $\mathrm{d} y / \mathrm{d} t=y(r-a \ln y)$ is equivalent to $N^{\prime}(t)=r N(t) \ln (K / N(t))$.

Eric Mockensturm

Numerade Educator

01:07

Problem 31

The equilibrium solution $y=c$ is classified as semistable if solutions on side of $y=c$ approach $y=c$ as $t \rightarrow \infty$ but on the other side of $y=c$, they move away from $y=c$ as $t \rightarrow \infty$. Use this definition to classify the equilibrium solutions (as asymptotically stable, semistable, or unstable) of the following differential equations.
(a) $y^{\prime}=y(y-2)^{2}$.
(b) $y^{\prime}=y^{2}(y-1)$.
(c) $y^{\prime}=y-\sqrt{y}$.
(d) $y^{\prime}=y^{2}\left(9-y^{2}\right)$.
(e) $y^{\prime}=1+y^{3}$.
(f) $y^{\prime}=y-y^{3}$.
(g) $y^{\prime}=y^{2}-y^{3}$.

Victoria Dollar

Numerade Educator

01:18

Problem 32

Consider the Malthus population model with $k=0.01,0.05,0.1,0.5$, and $1.0$ using $y_{0}=1$. Solve the model, plot the solution with these values, and compare the results. How does the value of $k$ affect the solution?

Lili Schantz

Numerade Educator

01:40

Problem 33

Consider the logistic equation with $r=0.01$, $0.05,0.1,0.5$, and $1.0$ using $y_{0}=1$ and $a=1$. Solve the model, plot the solution with these values, and compare the results. How does the value of $r$ affect the solution?

Lucas Gagne

Numerade Educator

04:07

Problem 34

(Tumor and Organism Growth) Ludwig von Bertalanffy (1901-1972) made valuable contributions to the study of organism growth, including the growth of tumors, based on the relationship between body size of the organism and the metabolic rate. He theorized that weight is directly proportional to volume and that the metabolic rate is proportional to surface area. This indicates that surface area is proportional to $V^{2 / 3}$ because $V$ is measured in cubic units and $S$ in square units. Therefore, Bertalanffy studied the IVP $\mathrm{d} V / \mathrm{d} t=a V^{2 / 3}-b V, V(0)=V_{0} .$
(a) Solve this IVP to show that volume is given by
$$
V(t)=\left[(a / b)-\left((a / b)-V_{0}^{1 / 3}\right) \mathrm{e}^{-b t / 3}\right]^{3} .
$$
(b) Find $\lim _{t \rightarrow \infty} V(t)$ and explain what the limit represents.
(c) This IVP is similar to that involving the logistic equation $\mathrm{d} V / \mathrm{d} t=a V^{2}-b V$, $V(0)=V_{0}$. Compare the limiting volume between the logistic equation and the model proposed by Bertalanffy in the cases when $a / b>1$ and when $a / b<1$.

Arwa Ali

Numerade Educator

View

Problem 35

(Harvesting) If we wish to model a population of size $P(t)$ at time $t$ and consider a constant harvest rate $h$ (like hunting, fishing, or disease), then we might modify the logistic equation and use the equation $P^{\prime}=r P-a P^{2}-h$ to model the population under consideration. Assume that $h \geq r^{2} /(4 a)$.
(a) Show that if $h \geq r^{2} /(4 a)$, a general solution of $P^{\prime}=r P-a P^{2}-h$ is $\begin{aligned} P(t)=& \frac{1}{2 a}\left[r+\sqrt{4 a h-r^{2}} \tan \right.\\ \times &\left.\left(\frac{1}{2 a}(C-a t) \sqrt{4 a h-r^{2}}\right)\right]. \end{aligned}$
(b) Suppose that for a certain species it is found that $r=0.03, a=0.0001, h=2.26$, and $C=-1501.85$. At what time will the species become extinct?
(c) If $r=0.03, a=0.0001$, and $P(0)=5.3$, graph $P(t)$ if $h=0,0.5,1.0,1.5,2.0,2.25$ and 2.5.
(d) What is the maximum allowable harvest rate to assure that the species survives?
(e) Generalize your result from (d). For arbitrary $a$ and $r$, what is the maximum allowable harvest rate that ensures survival of the species?

Susan Hallstrom

Numerade Educator

04:00

Problem 36

(Radiation Poisoning) Shortly after the Chernobyl accident in the Soviet Union in 1986, several nations reported that the level of ${ }^{131} \mathrm{I}$ in milk wasfive times that considered safe for human consumption. Make a table of the level of ${ }^{131} \mathrm{I}$ in milk as a multiple of that considered safe for human consumption for the first 3 weeks following the accident. After how long did the milk become safe for human consumption?

Nicholas Majtenyi

Numerade Educator

07:22

Problem 37

Consider a solution to the logistic equation with initial population $y_{0}$ where $0<y_{0}<r / a$. Show that this solution is concave up for $0<y<r /(2 a)$ and concave down for $r /(2 a)<y<r / a$. Hint:
Differentiate the right side of the logistic equation and set the result equal to 0 . Describe the behavior of $\mathrm{d} y / \mathrm{d} t$ based on this result.

Sirat Shah

Numerade Educator

04:45

Problem 38

From the early 1800 s to the mid- 1800 s, the passenger pigeon population was thriving. However, due to hunting, the population size was reduced dramatically by the late 1800 s. Unfortunately, the passenger pigeon requires a large number of cohorts to achieve successful reproduction. Having fallen below this level, the population size continued to decrease and the bird is now extinct. Which population model should be used to describe this situation?

Ryan Mcalister

Numerade Educator

04:27

Problem 39

(a) Find the solution to the IVP $\mathrm{d} y / \mathrm{d} t=-r(1-y / A) y, y(0)=y_{0}$, where $r$ and $A$ are positive constants. (b) If $y_{0}<A$, then determine $\lim _{t \rightarrow \infty} y(t)$. (c) If $y_{0}>A$, show that $\lim _{t \rightarrow \infty} y(t)=\infty$. (d) Show that if $y_{0}>A$, then $y(t)$ has a vertical asymptote at $t=(1 / r) \ln \left(y_{0} /\left(y_{0}-A\right)\right)$.

Bahar Tehranipoor

Numerade Educator

02:42

Problem 40

Solve the IVP $\mathrm{d} y / \mathrm{d} t=r(1-(1 / A) y)(1-(1 / B) y), y(0)=y_{0}$.

Shamshad Waris

Numerade Educator

03:05

Problem 41

(The Logistic Difference Equation) Given $x_{0}$, the logistic difference equation is
$$
x_{n+1}=r x_{n}\left(1-x_{n}\right) .
$$
Assume that $x_{0}=0.5$.
(a) If $r=3.83$, calculate $\left(n, x_{n}\right)$ for $n=1,2$, $\ldots, 50$, plot the resulting set of points and connect consecutive points with line segments. Why do you think this is called a "three-cycle?"
(b) Compute $\left(r, x_{n}(r)\right)$ for 250 equally spaced values of $r$ between $2.8$ and $4.0$ and $n=101, \ldots, 300$. Plot the resulting set of points. This famous image is called the "Pitchfork diagram."
(c) Repeat (b) for 250 equally spaced values of $r$ between $3.7$ and $4.0$ and $n=101, \ldots$, 300 .
(d) The logistic difference equation exhibits "chaos." Approximate the $r$-values between $2.8$ and $4.0$ for which the logistic difference equation exhibits chaos. Explain your reasoning. In your explanation, approximate those $r$-values that lead to a two-cycle, four-cycle, and so on.

Anish Wadhwa

Numerade Educator

04:00

Problem 42

(Growth in the Chemostat) The scaled equations for the growth of a population in a chemostat are
$$
\begin{gathered}
\frac{\mathrm{d} S}{\mathrm{~d} t}=1-S-\frac{m S}{a+S} x, \\
\frac{\mathrm{d} x}{\mathrm{~d} t}=x\left(\frac{m S}{a+S}-1\right), \\
\quad S(0) \geq 0, \quad x(0)>0,
\end{gathered}
$$
where $S(t)$ denotes the concentration of the nutrient at time $t$ for the organism with concentration $x(t)$ at time $t$.
(a) If $\Sigma=1-S-x$, show that $\Sigma^{\prime}=-\Sigma$. Then, system (3.6) can be written as
$$
\begin{aligned}
\frac{\mathrm{d} \Sigma}{\mathrm{d} t}=&-\Sigma \\
\frac{\mathrm{d} x}{\mathrm{~d} t}=& x\left(\frac{m(1-\Sigma-x)}{a+(1-\Sigma-x)}-1\right) \\
& \Sigma(0)>0, \quad x(0)>0
\end{aligned}
$$
Because $\Sigma(t)=\Sigma(0) \mathrm{e}^{-t}, \lim _{t \rightarrow \infty} \Sigma(t)=0$
so system (3.7) can be rewritten as the single first-order equation
$$
\begin{aligned}
\frac{\mathrm{d} x}{\mathrm{~d} t} &=x\left(\frac{m(1-x)}{a+(1-x)}-1\right) & \text { or } \\
\frac{\mathrm{d} x}{\mathrm{~d} t} &=x\left[\frac{m(1-x)}{1+a-x}-1\right], & 0 \leq x \leq 1,
\end{aligned}
$$
where $x(0)>0$.
(b) Find the equilibrium solutions of Equation (3.8).
(c) Write the nontrivial solutions found in (a) in the form $1-\lambda$. ( $\lambda$ is called the break-even concentration.)
(d) Find conditions on $\lambda$ so that a nontrivial equilibrium solution exists. Confirm your result by graphing various solutions for various values of $m$ and $a$. Illustrate that if $m$ is sufficiently small, the organism becomes extinct, regardless of the initial conditions. On the other hand, if a nontrivial equilibrium solution exists, all nontrivial solutions tend to it.

Carson Merrill

Numerade Educator