Elementary Differential Equations and Boundary Value Problem

William E. Boyce, Richard C. DiPrima

Chapter 1

Introduction - all with Video Answers

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Section 1

Some Basic Mathematical Models; Direction Fields

Problem 1

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$, If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency.
$$
y^{\prime}=3-2 y
$$

Aidan Mcnabb

Aidan Mcnabb

Numerade Educator

Problem 2

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$, If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency.
$$
y^{\prime}=2 y-3
$$

Aidan Mcnabb

Numerade Educator

Problem 3

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$, If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency.
$$
y^{\prime}=3+2 y
$$

Aidan Mcnabb

Numerade Educator

Problem 4

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$, If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency.
$$
y^{\prime}=-1-2 y
$$

Aidan Mcnabb

Numerade Educator

Problem 5

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$, If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency.
$$
y^{\prime}=1+2 y
$$

Aidan Mcnabb

Numerade Educator

Problem 6

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$, If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency.
$$
y^{\prime}=y+2
$$

Aidan Mcnabb

Numerade Educator

Problem 7

In cach of Problems 7 through 10 write down a differential equation of the form $d y / d t=a y+b$ whose solutions have the required bchavior as $t \rightarrow \infty$.

All solutions approach $y=3$

Aidan Mcnabb

Numerade Educator

Problem 8

In cach of Problems 7 through 10 write down a differential equation of the form $d y / d t=a y+b$ whose solutions have the required bchavior as $t \rightarrow \infty$.

All solutions approach $y=2 / 3$

Aidan Mcnabb

Numerade Educator

Problem 9

In cach of Problems 7 through 10 write down a differential equation of the form $d y / d t=a y+b$ whose solutions have the required bchavior as $t \rightarrow \infty$.

All other solutions diverge from $y=2$

Aidan Mcnabb

Numerade Educator

Problem 10

In cach of Problems 7 through 10 write down a differential equation of the form $d y / d t=a y+b$ whose solutions have the required bchavior as $t \rightarrow \infty$.

All other solutions diverge from $y=1 / 3$

Aidan Mcnabb

Numerade Educator

Problem 11

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0$, describe this dependency. Note that in these problems the equations are not of the form $y^{\prime}=a y+b$ and the behavior of their solutions is somewhat more complicated than for the equations in the text.
$$
y^{\prime}=y(4-y)
$$

Aidan Mcnabb

Numerade Educator

Problem 12

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0$, describe this dependency. Note that in these problems the equations are not of the form $y^{\prime}=a y+b$ and the behavior of their solutions is somewhat more complicated than for the equations in the text.
$$
y^{\prime}=-y(5-y)
$$

Aidan Mcnabb

Numerade Educator

Problem 13

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0$, describe this dependency. Note that in these problems the equations are not of the form $y^{\prime}=a y+b$ and the behavior of their solutions is somewhat more complicated than for the equations in the text.
$$
y^{\prime}=y^{2}
$$

Madi Sousa

Numerade Educator

Problem 14

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0$, describe this dependency. Note that in these problems the equations are not of the form $y^{\prime}=a y+b$ and the behavior of their solutions is somewhat more complicated than for the equations in the text.
$$
y^{\prime}=y(y-2)^{2}
$$

Aidan Mcnabb

Numerade Educator

Problem 15

A pond initially contains $1,000,000$ gal of water and an unknown amount of an undesirable
chemical. Water containing 0.01 gram of this chemical per gallon flows into the pond at a rate of $300 \mathrm{gal} / \mathrm{min}$. The mixture flows out at the same rate so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond.
(a) Write a differential equation whose solution is the amount of chemical in the pond at
any time
(b) How much of the chemical will be in the pond after a very long time? Does this limiting
amount depend on the amount that was present initially?

Chris Trentman

Numerade Educator

Problem 16

A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential
equation for the volume of the raindrop as a function of time.

Carson Merrill

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Problem 17

A certain drug is being administered intravenously to a hospital patient, Fluid containing $5 \mathrm{mg} / \mathrm{cm}^{3}$ of the drug enters the patient's bloodstream at a rate of $100 \mathrm{cm}^{3} \mathrm{hr}$. The drug is absorbed by body tissues or otherwise leaves the bloodstream at a rate proportional to the amount present, with a rate constant of $0.4(\mathrm{hr})^{-1}$.
(a) Assuming that the drug is always uniformly distributed throughout the bloodstream,
write a differential equation for the amount of the drug that is present in the bloodstream,
at any time.
(b) How much of the drug is present in the bloodstream after a long time?

Susan Hallstrom

Susan Hallstrom

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Problem 18

For small, slowly falling objects the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects it is more accurate to assume that the drag force is proportional to the square of the velocity."
(a) Write a differential equation for the velocity of a falling object of mass $m$ if the drag force is proportional to the square of the velocity.
(b) Determine the limiting velocity after a long time.
(c) If $m=10 \mathrm{kg}$, find the drag cocficient so that the limiting velocity is $49 \mathrm{m} / \mathrm{sec}$.
(d) Using the data in part (c), draw a direction field and compare it with Figure $1.13 .$

James Kiss

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Problem 19

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency. Note the right sides of these equations depend on $t$ as well as $y$, therefore their solutions can exhibit more complicated behavior than those in the text.
$$
y^{\prime}=-2+t-y
$$

Aidan Mcnabb

Numerade Educator

Problem 20

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency. Note the right sides of these equations depend on $t$ as well as $y$, therefore their solutions can exhibit more complicated behavior than those in the text.
$$
y^{\prime}=t e^{-2 t}-2 y
$$

Aidan Mcnabb

Numerade Educator

Problem 21

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency. Note the right sides of these equations depend on $t$ as well as $y$, therefore their solutions can exhibit more complicated behavior than those in the text.
$$
y^{\prime}=e^{-t}+y
$$

Aidan Mcnabb

Numerade Educator

Problem 22

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency. Note the right sides of these equations depend on $t$ as well as $y$, therefore their solutions can exhibit more complicated behavior than those in the text.
$$
y^{\prime}=t+2 y
$$

Aidan Mcnabb

Numerade Educator

Problem 23

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency. Note the right sides of these equations depend on $t$ as well as $y$, therefore their solutions can exhibit more complicated behavior than those in the text.
$$
y^{\prime}=3 \sin t+1+y
$$

Aidan Mcnabb

Numerade Educator

Problem 24

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency. Note the right sides of these equations depend on $t$ as well as $y$, therefore their solutions can exhibit more complicated behavior than those in the text.
$$
y^{\prime}=2 t-1-y^{2}
$$

Aidan Mcnabb

Numerade Educator

Problem 25

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency. Note the right sides of these equations depend on $t$ as well as $y$, therefore their solutions can exhibit more complicated behavior than those in the text.
$$
y^{\prime}=-(2 t+y) / 2 y
$$

Aidan Mcnabb

Numerade Educator

Problem 26

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of $y$ as $t \rightarrow \infty$. If this behavior depends on the initial value of $y$ at $t=0,$ describe this dependency. Note the right sides of these equations depend on $t$ as well as $y$, therefore their solutions can exhibit more complicated behavior than those in the text.
$$
y^{\prime}=y^{3} / 6-y-t^{2} / 3
$$

IL

Iris L.

Numerade Educator