Thomas Calculus

George B. Thomas, Jr.

Chapter 9

First-Order Differential Equations - all with Video Answers

Educators

KM

+ 3 more educators

Section 1

Solutions, Slope Fields, and Euler's Method

Problem 1

In Exercises $1-4,$ match the differential equations with their slope
fields, graphed here.
$$
y^{\prime}=x+y
$$

Cyrielle Lorio

Cyrielle Lorio

Numerade Educator

Problem 2

In Exercises $1-4,$ match the differential equations with their slope
fields, graphed here.
$$
y^{\prime}=y+1
$$

Cyrielle Lorio

Numerade Educator

Problem 3

In Exercises $1-4,$ match the differential equations with their slope
fields, graphed here.
$$
y^{\prime}=-\frac{x}{y}
$$

Cyrielle Lorio

Numerade Educator

Problem 4

In Exercises $1-4,$ match the differential equations with their slope
fields, graphed here.
$$
y^{\prime}=y^{2}-x^{2}
$$

Cyrielle Lorio

Numerade Educator

Problem 5

In Exercises 5 and $6,$ copy the slope fields and sketch in some of the
solution curves.
$$
y^{\prime}=(y+2)(y-2)
$$

Bobby Barnes

University of North Texas

Problem 6

In Exercises 5 and $6,$ copy the slope fields and sketch in some of the
solution curves.
$$
y^{\prime}=y(y+1)(y-1)
$$

Bobby Barnes

University of North Texas

Problem 7

In Exercises $7-10$ , write an equivalent first-order differential equation
and initial condition for $y .$
$$
y=-1+\int_{1}^{x}(t-y(t)) d t
$$

KL

Kang Lu

Numerade Educator

Problem 8

In Exercises $7-10$ , write an equivalent first-order differential equation
and initial condition for $y .$
$$
y=\int_{1}^{x} \frac{1}{t} d t
$$

KL

Kang Lu

Numerade Educator

Problem 9

In Exercises $7-10$ , write an equivalent first-order differential equation
and initial condition for $y .$
$$
y=2-\int_{0}^{x}(1+y(t)) \sin t d t
$$

KL

Kang Lu

Numerade Educator

Problem 10

In Exercises $7-10,$ write an equivalent first-order differential equation
and initial condition for $y .$
$$
y=1+\int_{0}^{x} y(t) d t
$$

KL

Kang Lu

Numerade Educator

Problem 11

In Exercises $11-16,$ use Euler's method to calculate the first three
approximations to the given initial value problem for the specified
increment size. Calculate the exact solution and investigate the accuracy
of your approximations. Round your results to four decimal places.
$$
y^{\prime}=1-\frac{y}{x}, \quad y(2)=-1, \quad d x=0.5
$$

Audrey Fong

Numerade Educator

Problem 12

In Exercises $11-16,$ use Euler's method to calculate the first three
approximations to the given initial value problem for the specified
increment size. Calculate the exact solution and investigate the accuracy
of your approximations. Round your results to four decimal places.
$$
y^{\prime}=x(1-y), \quad y(1)=0, \quad d x=0.2
$$

Audrey Fong

Numerade Educator

Problem 13

In Exercises $11-16,$ use Euler's method to calculate the first three
approximations to the given initial value problem for the specified
increment size. Calculate the exact solution and investigate the accuracy
of your approximations. Round your results to four decimal places.
$$
y^{\prime}=2 x y+2 y, \quad y(0)=3, \quad d x=0.2
$$

Audrey Fong

Numerade Educator

Problem 14

In Exercises $11-16,$ use Euler's method to calculate the first three
approximations to the given initial value problem for the specified
increment size. Calculate the exact solution and investigate the accuracy
of your approximations. Round your results to four decimal places.
$$
y^{\prime}=y^{2}(1+2 x), \quad y(-1)=1, \quad d x=0.5
$$

Audrey Fong

Numerade Educator

Problem 15

In Exercises $11-16,$ use Euler's method to calculate the first three
approximations to the given initial value problem for the specified
increment size. Calculate the exact solution and investigate the accuracy
of your approximations. Round your results to four decimal places.
$$
y^{\prime}=2 x e^{x^{2}}, \quad y(0)=2, \quad d x=0.1
$$

Audrey Fong

Numerade Educator

Problem 16

In Exercises $11-16,$ use Euler's method to calculate the first three
approximations to the given initial value problem for the specified
increment size. Calculate the exact solution and investigate the accuracy
of your approximations. Round your results to four decimal places.
$$
y^{\prime}=y e^{x}, \quad y(0)=2, \quad d x=0.5
$$

Audrey Fong

Numerade Educator

Problem 17

Use the Euler method with $d x=0.2$ to estimate $y(1)$ if $y^{\prime}=y$
and $y(0)=1 .$ What is the exact value of $y(1) ?$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

Problem 18

Use the Euler method with $d x=0.2$ to estimate $y(2)$ if $y^{\prime}=y / x$
and $y(1)=2 .$ What is the exact value of $y(2) ?$

Clarissa Noh

Numerade Educator

Problem 19

Use the Euler method with $d x=0.5$ to estimate $y(5)$ if $y^{\prime}=$
$y^{2} / \sqrt{x}$ and $y(1)=-1 .$ What is the exact value of $y(5) ?$

Clarissa Noh

Numerade Educator

Problem 20

Use the Euler method with $d x=0.5$ to estimate $y(5)$ if $y^{\prime}=$
$y^{2} / \sqrt{x}$ and $y(1)=-1 .$ What is the exact value of $y(5) ?$

Clarissa Noh

Numerade Educator

Problem 21

Show that the solution of the initial value problem
is
$y^{\prime}=x+y, \quad y\left(x_{0}\right)=y_{0}$
$y=-1-x+\left(1+x_{0}+y_{0}\right) e^{x-x_{0}}$

Clarissa Noh

Numerade Educator

Problem 22

What integral equation is equivalent to the initial value problem
$y^{\prime}=f(x), y\left(x_{0}\right)=y_{0} ?$

Rahul Kumar

Numerade Educator

Problem 23

In Exercises $23-28,$ obtain a slope field and add to it graphs of the
solution curves passing through the given points.
$$
\begin{array}{l}{y^{\prime}=y \text { with }} \\ {\text { a. }(0,1) \quad \text { b. }(0,2) \quad \text { c. }(0,-1)}\end{array}
$$

Andrija Isakov

Numerade Educator

Problem 24

In Exercises $23-28,$ obtain a slope field and add to it graphs of the
solution curves passing through the given points.
$$
\begin{array}{l}{y^{\prime}=2(y-4) \text { with }} \\ {\text { a. }(0,1) \quad \text { b. }(0,4)}\end{array} \quad \text { c. }(0,5)
$$

Andrija Isakov

Numerade Educator

Problem 25

In Exercises $23-28,$ obtain a slope field and add to it graphs of the
solution curves passing through the given points.
$$
\begin{array}{l}{y^{\prime}=y(x+y) \text { with }} \\ {\text { a. }(0,1) \quad \text { b. }(0,-2) \quad \text { c. }(0,1 / 4) \quad \text { d. }(-1,-1)}\end{array}
$$

Andrija Isakov

Numerade Educator

Problem 26

In Exercises $23-28,$ obtain a slope field and add to it graphs of the
solution curves passing through the given points.
$$
\begin{array}{l}{y^{\prime}=y^{2} \text { with }} \\ {\text { a. }(0,1) \quad \text { b. }(0,2)}\end{array} \quad \text { c. }(0,-1) \quad \text { d. }(0,0)
$$

Andrija Isakov

Numerade Educator

Problem 27

In Exercises $23-28,$ obtain a slope field and add to it graphs of the
solution curves passing through the given points.
$$
\begin{array}{ll}{y^{\prime}=(y-1)(x+2) \text { with }} \\ {\text { a. }(0,-1) \quad \text { b. }(0,1)} & {\text { c. }(0,3)}\end{array} \quad \text { d. }(1,-1)
$$

Andrija Isakov

Numerade Educator

Problem 28

In Exercises $23-28,$ obtain a slope field and add to it graphs of the
solution curves passing through the given points.
$$
\begin{array}{l}{y^{\prime}=\frac{x y}{x^{2}+4} \text { with }} \\ {\text { a. }(0,2) \quad \text { b. }(0,-6) \quad \text { c. }(-2 \sqrt{3},-4)}\end{array}
$$

Andrija Isakov

Numerade Educator

Problem 29

In Exercises 29 and $30,$ obtain a slope field and graph the particular
solution over the specified interval. Use your CAS DE solver to find
the general solution of the differential equation.
$$
\begin{array}{l}{\text { A logistic equation } y^{\prime}=y(2-y), y(0)=1 / 2 ; 0 \leq x \leq 4} \\ {0 \leq y \leq 3}\end{array}
$$

Andrija Isakov

Numerade Educator

Problem 30

In Exercises 29 and $30,$ obtain a slope field and graph the particular
solution over the specified interval. Use your CAS DE solver to find
the general solution of the differential equation.
$$
y^{\prime}=(\sin x)(\sin y), \quad y(0)=2 ; \quad-6 \leq x \leq 6, \quad-6 \leq y \leq 6
$$

Andrija Isakov

Numerade Educator

Problem 31

Exercises 31 and 32 have no explicit solution in terms of elementary
functions. Use a CAS to explore graphically each of the differential
equations.
$$
y^{\prime}=\cos (2 x-y), \quad y(0)=2 ; \quad 0 \leq x \leq 5, \quad 0 \leq y \leq 5
$$

Stephen Hobbs

Numerade Educator

Problem 32

Exercises 31 and 32 have no explicit solution in terms of elementary
functions. Use a CAS to explore graphically each of the differential
equations.
\begin{equation}
\begin{array}{l}{\text { A Gompertz equation } y^{\prime}=y(1 / 2-\ln y), \quad y(0)=1 / 3} \\ {0 \leq x \leq 4, \quad 0 \leq y \leq 3}\end{array}
\end{equation}

Stephen Hobbs

Numerade Educator

Problem 33

$\begin{equation}
\begin{array}{l}{\text { Use a CAS to find the solutions of } y^{\prime}+y=f(x) \text { subject to the }} \\ {\text { initial condition } y(0)=0, \text { if } f(x) \text { is }} \\ {\text { a. } 2 x \text { b. } \sin 2 x} {\text { c. } 3 e^{x / 2}} \\ {\text { Graph all four solutions over the interval }-2 \leq x \leq 6 \text { to com- }} \\ {\text { pare the results. }}\end{array}
\end{equation}$

Regina Hays

Numerade Educator

Problem 34

a. Use a CAS to plot the slope field of the differential equation
$$
y^{\prime}=\frac{3 x^{2}+4 x+2}{2(y-1)}
$$
$$
-3 \leq x \leq 3 \text { and }-3 \leq y \leq 3
$$
b. Separate the variables and use a CAS integrator to find the
general solution in implicit form.
c. Using a CAS implicit function grapher, plot solution curves
for the arbitrary constant values $C=-6,-4,-2,0,2,4,6$ .
d. Find and graph the solution that satisfies the initial condition
$y(0)=-1 .$

Regina Hays

Numerade Educator

Problem 35

In Exercises $35-38$ use Euler's method with the specified step size to
estimate the value of the solution at the given point $x^{*} .$ Find the value
of the exact solution at $x^{*} .$
$$
y^{\prime}=2 x e^{x^{2}}, \quad y(0)=2, \quad d x=0.1, \quad x^{*}=1
$$

Victor Salazar

Numerade Educator

Problem 36

In Exercises $35-38$ use Euler's method with the specified step size to
estimate the value of the solution at the given point $x^{*} .$ Find the value
of the exact solution at $x^{*} .
$$
y^{\prime}=2 y^{2}(x-1), \quad y(2)=-1 / 2, \quad d x=0.1, x^{*}=3
$$

Victor Salazar

Numerade Educator

Problem 37

In Exercises $35-38$ use Euler's method with the specified step size to
estimate the value of the solution at the given point $x^{*} .$ Find the value
of the exact solution at $x^{*} .
$$
y^{\prime}=\sqrt{x} / y, \quad y>0, \quad y(0)=1, \quad d x=0.1, \quad x^{*}=1
$$

Victor Salazar

Numerade Educator

Problem 38

In Exercises $35-38$ use Euler's method with the specified step size to
estimate the value of the solution at the given point $x^{*} .$ Find the value
of the exact solution at $x^{*} .
$$
y^{\prime}=1+y^{2}, \quad y(0)=0, \quad d x=0.1, \quad x^{*}=1
$$

Victor Salazar

Numerade Educator

Problem 39

Use a CAS to explore graphically each of the differential equations in
Exercises $39-42 .$ Perform the following steps to help with your explo-
rations.
a. Plot a slope field for the differential equation in the given
b. Find the general solution of the differential equation using
your CAS DE solver.
c. Graph the solutions for the values of the arbitrary constant
$C=-2,-1,0,1,2$ superimposed on your slope field plot.
d. Find and graph the solution that satisfies the specified initial
condition over the interval $[0, b]$ .
e. Find the Euler numerical approximation to the solution of the
initial value problem with 4 subintervals of the $x$ -interval and
plot the Euler approximation superimposed on the graph produced in part (d).
f. Repeat part (e) for $8,16,$ and 32 subintervals. Plot these three
Euler approximations superimposed on the graph from part (e).
g. Find the error $(y($ exact $)-y($ Exact $)$ at the specified point
$x=b$ for each of your four Euler approximations. Discuss
the improvement in the percentage error.
$$
\begin{array}{l}{y^{\prime}=x+y, \quad y(0)=-7 / 10 ; \quad-4 \leq x \leq 4, \quad-4 \leq y \leq 4} \\ {b=1}\end{array}
$$

Regina Hays

Numerade Educator

Problem 40

Use a CAS to explore graphically each of the differential equations in
Exercises $39-42 .$ Perform the following steps to help with your explo-
rations.
a. Plot a slope field for the differential equation in the given
b. Find the general solution of the differential equation using
your CAS DE solver.
c. Graph the solutions for the values of the arbitrary constant
$C=-2,-1,0,1,2$ superimposed on your slope field plot.
d. Find and graph the solution that satisfies the specified initial
condition over the interval $[0, b]$ .
e. Find the Euler numerical approximation to the solution of the
initial value problem with 4 subintervals of the $x$ -interval and
plot the Euler approximation superimposed on the graph produced in part (d).
f. Repeat part (e) for $8,16,$ and 32 subintervals. Plot these three
Euler approximations superimposed on the graph from part (e).
g. Find the error $(y($ exact $)-y($ Exact $)$ at the specified point
$x=b$ for each of your four Euler approximations. Discuss
the improvement in the percentage error.
$$
y^{\prime}=-x / y, \quad y(0)=2 ; \quad-3 \leq x \leq 3, \quad-3 \leq y \leq 3 ; b=2
$$

Regina Hays

Numerade Educator

Problem 41

Use a CAS to explore graphically each of the differential equations in
Exercises $39-42 .$ Perform the following steps to help with your explo-
rations.
a. Plot a slope field for the differential equation in the given
b. Find the general solution of the differential equation using
your CAS DE solver.
c. Graph the solutions for the values of the arbitrary constant
$C=-2,-1,0,1,2$ superimposed on your slope field plot.
d. Find and graph the solution that satisfies the specified initial
condition over the interval $[0, b]$ .
e. Find the Euler numerical approximation to the solution of the
initial value problem with 4 subintervals of the $x$ -interval and
plot the Euler approximation superimposed on the graph produced in part (d).
f. Repeat part (e) for $8,16,$ and 32 subintervals. Plot these three
Euler approximations superimposed on the graph from part (e).
g. Find the error $(y($ exact $)-y($ Exact $)$ at the specified point
$x=b$ for each of your four Euler approximations. Discuss
the improvement in the percentage error.
$$
y^{\prime}=y(2-y), \quad y(0)=1 / 2 ; \quad 0 \leq x \leq 4,0 \leq y \leq 3 ; b=3
$$

Regina Hays

Numerade Educator

Problem 42

Use a CAS to explore graphically each of the differential equations in
Exercises $39-42 .$ Perform the following steps to help with your explo-
rations.
a. Plot a slope field for the differential equation in the given
b. Find the general solution of the differential equation using
your CAS DE solver.
c. Graph the solutions for the values of the arbitrary constant
$C=-2,-1,0,1,2$ superimposed on your slope field plot.
d. Find and graph the solution that satisfies the specified initial
condition over the interval $[0, b]$ .
e. Find the Euler numerical approximation to the solution of the
initial value problem with 4 subintervals of the $x$ -interval and
plot the Euler approximation superimposed on the graph produced in part (d).
f. Repeat part (e) for $8,16,$ and 32 subintervals. Plot these three
Euler approximations superimposed on the graph from part (e).
g. Find the error $(y($ exact $)-y($ Exact $)$ at the specified point
$x=b$ for each of your four Euler approximations. Discuss
the improvement in the percentage error.
$$
\begin{array}{l}{y^{\prime}=(\sin x)(\sin y), \quad y(0)=2 ; \quad-6 \leq x \leq 6, \quad-6 \leq y \leq 6} \\ {b=3 \pi / 2}\end{array}
$$

Regina Hays

Numerade Educator