University Calculus: Early Transcendentals

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 16

First-Order Differential Equations - all with Video Answers

Educators

Section 1

Solutions, Slope Fields, and Euler’s Method

Problem 1

Match the differential equations with their slope fields, graphed here. (GRAPHS CANNOT COPY)
$$y^{\prime}=x+y$$

Regina H.

Regina H.

Numerade Educator

Problem 2

Match the differential equations with their slope fields, graphed here. (GRAPHS CANNOT COPY)
$$y^{\prime}=y+1$$

Regina H.

Numerade Educator

Problem 3

Match the differential equations with their slope fields, graphed here. (GRAPHS CANNOT COPY)
$$y^{\prime}=-\frac{x}{y}$$

Regina H.

Numerade Educator

Problem 4

Match the differential equations with their slope fields, graphed here. (GRAPHS CANNOT COPY)
$$y^{\prime}=y^{2}-x^{2}$$

Regina H.

Numerade Educator

Problem 5

Copy the slope fields, and sketch in some of the solution curves.
$$y^{\prime}=(y+2)(y-2)$$
(GRAPH CANNOT COPY)

Regina H.

Numerade Educator

Problem 6

Copy the slope fields, and sketch in some of the solution curves.
$$y^{\prime}=y(y+1)(y-1)$$
(GRAPH CANNOT COPY)

Regina H.

Numerade Educator

Problem 7

Write an equivalent first-order differential equation and initial condition for $y.$
$$y=-1+\int_{1}^{x}(t-y(t)) d t$$

Clarissa N.

Numerade Educator

Problem 8

Write an equivalent first-order differential equation and initial condition for $y.$
$$y=\int_{1}^{x} \frac{1}{t} d t$$

Brittany K.

Numerade Educator

Problem 9

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

Regina H.

Numerade Educator

Problem 10

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

Brittany K.

Numerade Educator

Problem 11

Write an equivalent first-order differential equation and initial condition for $y.$
$$y=x+4+\int_{-2}^{x} t e^{y(n)} d t$$

Regina H.

Numerade Educator

Problem 12

Write an equivalent first-order differential equation and initial condition for $y.$
$$y=\ln x+\int_{x}^{e} \sqrt{t^{2}+(y(t))^{2}} d t$$

Brittany K.

Numerade Educator

Problem 13

Consider the differential equation $y^{\prime}=f(y)$ and the given graph of $f .$ Make a rough sketch of a direction field for each differential equation.
(GRAPH CANNOT COPY)

Regina H.

Numerade Educator

Problem 14

Consider the differential equation $y^{\prime}=f(y)$ and the given graph of $f .$ Make a rough sketch of a direction field for each differential equation.
(GRAPH CANNOT COPY)

Regina H.

Numerade Educator

Problem 15

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. $$y^{\prime}=\frac{2 y}{x}, \quad y(1)=-1, \quad d x=0.5$$

Clarissa N.

Numerade Educator

Problem 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. $$y^{\prime}=x(1-y), \quad y(1)=0, \quad d x=0.2$$

Brittany K.

Numerade Educator

Problem 17

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. $$y^{\prime}=2 x y+2 y, \quad y(0)=3, \quad d x=0.2$$

Clarissa N.

Numerade Educator

Problem 18

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. $$y^{\prime}=y^{2}(1+2 x), \quad y(-1)=1, \quad d x=0.5$$

Brittany K.

Numerade Educator

Problem 19

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. $$y^{\prime}=2 x e^{x^{2}}, \quad y(0)=2, \quad d x=0.1$$

Clarissa N.

Numerade Educator

Problem 20

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. $$y^{\prime}=y e^{x}, \quad y(0)=2, \quad d x=0.5$$

Brittany K.

Numerade Educator

Problem 21

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 $y(1) ?$

Clarissa N.

Numerade Educator

Problem 22

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 $y(2) ?$

Brittany K.

Numerade Educator

Problem 23

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 N.

Numerade Educator

Problem 24

Use the Euler method with $d x=1 / 3$ to estimate $y(2)$ if $y^{\prime}=$ $x \sin y$ and $y(0)=1 .$ What is the exact value of $y(2) ?$

Brittany K.

Numerade Educator

Problem 25

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

Clarissa N.

Numerade Educator

Problem 26

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

Brittany K.

Numerade Educator

Problem 27

Obtain a slope field, and add to it graphs of the solution curves passing through the given points.
$y^{\prime}=y$ with
a. (0,1)
b. (0,2)
c. (0,-1)

Regina H.

Numerade Educator

Problem 28

Obtain a slope field, and add to it graphs of the solution curves passing through the given points.
$y^{\prime}=2(y-4)$ with
a. (0,1)
b. (0,4)
c. (0,5)

Regina H.

Numerade Educator

Problem 29

Obtain a slope field, and add to it graphs of the solution curves passing through the given points.
$y^{\prime}=y(x+y)$ with
a. (0,1)
b. (0,-2)
c. $(0,1 / 4)$
d. (-1,-1)

Regina H.

Numerade Educator

Problem 30

Obtain a slope field, and add to it graphs of the solution curves passing through the given points.
$y^{\prime}=y^{2}$ with
a. (0,1)
b. (0,2)$\quad$ c. (0,-1)
d. (0,0)

Regina H.

Numerade Educator

Problem 31

Obtain a slope field, and add to it graphs of the solution curves passing through the given points.
$y^{\prime}=(y-1)(x+2)$ with
a. (0,-1)
b. (0,1)$\quad$ c. (0,3)
d. (1,-1)

Regina H.

Numerade Educator

Problem 32

Obtain a slope field, and add to it graphs of the solution curves passing through the given points.
$y^{\prime}=\frac{x y}{x^{2}+4}$ with
a. (0,2)
b. (0,-6)$\quad$ c. $(-2 \sqrt{3},-4)$

Regina H.

Numerade Educator

Problem 33

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.
A logistic equation $y^{\prime}=y(2-y), y(0)=1 / 2 ; 0 \leq x \leq 4$
$0 \leq y \leq 3$

Regina H.

Numerade Educator

Problem 34

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.
$y^{\prime}=(\sin x)(\sin y), \quad y(0)=2 ; \quad-6 \leq x \leq 6, \quad-6 \leq y \leq 6$

Regina H.

Numerade Educator

Problem 35

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

Clarissa N.

Numerade Educator

Problem 36

Have no explicit solution in terms of elementary functions. Use a CAS to explore graphically each of the differential equations.
A Gompertz equation $y^{\prime}=y(1 / 2-\ln y), \quad y(0)=1 / 3$
$0 \leq x \leq 4, \quad 0 \leq y \leq 3$

Brittany K.

Numerade Educator

Problem 37

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

Regina H.

Numerade Educator

Problem 38

a. Use a CAS to plot the slope field of the differential equation $$y^{\prime}=\frac{3 x^{2}+4 x+2}{2(y-1)}$$ over the region $-3 \leq x \leq 3$ 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 H.

Numerade Educator

Problem 39

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 $x^{*}.$
$$y^{\prime}=2 x e^{x^{2}}, \quad y(0)=2, \quad d x=0.1, \quad x^{*}=1$$

Clarissa N.

Numerade Educator

Problem 40

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 $x^{*}.$
$$y^{\prime}=2 y^{2}(x-1), \quad y(2)=-1 / 2, \quad d x=0.1, \quad x^{*}=3$$

Brittany K.

Numerade Educator

Problem 41

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 $x^{*}.$
$$y^{\prime}=\sqrt{x} / y, \quad y>0, \quad y(0)=1, \quad d x=0.1, \quad x^{*}=1$$

Regina H.

Numerade Educator

Problem 42

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 $x^{*}.$
$$y^{\prime}=1+y^{2}, \quad y(0)=0, \quad d x=0.1, \quad x^{*}=1$$

Brittany K.

Numerade Educator

Problem 43

Use a CAS to explore graphically each of the differential equations. Perform the following steps to help with your explorations.
a. Plot a slope field for the differential equation in the given $x y$ -window.
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) $-y$ (Euler)) at the specified point $x=b$ for each of your four Euler approximations. Discuss the improvement in the percentage error.
$$\begin{aligned}&y^{\prime}=x+y, \quad y(0)=-7 / 10 ; \quad-4 \leq x \leq 4, \quad-4 \leq y \leq 4\\&b=1\end{aligned}$$

Regina H.

Numerade Educator

Problem 44

Use a CAS to explore graphically each of the differential equations. Perform the following steps to help with your explorations.
a. Plot a slope field for the differential equation in the given $x y$ -window.
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) $-y$ (Euler)) at the specified point $x=b$ for each of your four Euler approximations. Discuss the improvement in the percentage error.
$$\begin{aligned}&y^{\prime}=-x / y, \quad y(0)=2 ; \quad-3 \leq x \leq 3, \quad-3 \leq y \leq 3\\ &b=2\end{aligned}$$

Regina H.

Numerade Educator

Problem 45

Use a CAS to explore graphically each of the differential equations. Perform the following steps to help with your explorations.
a. Plot a slope field for the differential equation in the given $x y$ -window.
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) $-y$ (Euler)) at the specified point $x=b$ for each of your four Euler approximations. Discuss the improvement in the percentage error.
$$\begin{aligned}&y^{\prime}=y(2-y), \quad y(0)=1 / 2 ; \quad 0 \leq x \leq 4,0 \leq y \leq 3\\&b=3\end{aligned}$$

Regina H.

Numerade Educator

Problem 46

Use a CAS to explore graphically each of the differential equations. Perform the following steps to help with your explorations.
a. Plot a slope field for the differential equation in the given $x y$ -window.
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) $-y$ (Euler)) 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 H.

Numerade Educator