• Home
  • Textbooks
  • Thomas Calculus
  • First-Order Differential Equations

Thomas Calculus

George B. Thomas Jr.

Chapter 9

First-Order Differential Equations - all with Video Answers

Educators

Section 1

Solutions Slope Fields and Eulers Method

00:33

Problem 1

Match the differential equations with their slope fields, graphed here.
$y^{\prime}=x+y$

Clarissa Noh
Clarissa Noh
Numerade Educator
02:21

Problem 2

Match the differential equations with their slope fields, graphed here.
$y^{\prime}=y+1$

Brittany Knowlton
Brittany Knowlton
Numerade Educator
00:24

Problem 3

Match the differential equations with their slope fields, graphed here.
$y^{\prime}=-\frac{x}{y}$

Clarissa Noh
Clarissa Noh
Numerade Educator
02:32

Problem 4

Match the differential equations with their slope fields, graphed here.
$y^{\prime}=y^{2}-x^{2}$

Brittany Knowlton
Brittany Knowlton
Numerade Educator
00:37

Problem 5

Copy the slope fields and sketch in some of the solution curves.
$y^{\prime}=(y+2)(y-2)$

Clarissa Noh
Clarissa Noh
Numerade Educator
01:25

Problem 6

Copy the slope fields and sketch in some of the solution curves.
$y^{\prime}=y(y+1)(y-1)$

Brittany Knowlton
Brittany Knowlton
Numerade Educator
01:00

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 Noh
Clarissa Noh
Numerade Educator
01:26

Problem 8

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

Brittany Knowlton
Brittany Knowlton
Numerade Educator
01:17

Problem 9

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

Clarissa Noh
Clarissa Noh
Numerade Educator
01:34

Problem 10

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

Brittany Knowlton
Brittany Knowlton
Numerade Educator
00:51

Problem 11

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

Clarissa Noh
Clarissa Noh
Numerade Educator
02:18

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 Knowlton
Brittany Knowlton
Numerade Educator
01:59

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.

Clarissa Noh
Clarissa Noh
Numerade Educator
03:46

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.

Brittany Knowlton
Brittany Knowlton
Numerade Educator
02:49

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 Noh
Clarissa Noh
Numerade Educator
08:30

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 Knowlton
Brittany Knowlton
Numerade Educator
04:24

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 Noh
Clarissa Noh
Numerade Educator
06:44

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 Knowlton
Brittany Knowlton
Numerade Educator
03:25

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 Noh
Clarissa Noh
Numerade Educator
06:55

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 Knowlton
Brittany Knowlton
Numerade Educator
02:36

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 Noh
Clarissa Noh
Numerade Educator
06:59

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 Knowlton
Brittany Knowlton
Numerade Educator
03:23

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 Noh
Clarissa Noh
Numerade Educator
09:39

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 Knowlton
Brittany Knowlton
Numerade Educator
01:52

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 Noh
Clarissa Noh
Numerade Educator
01:59

Problem 26

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

Brittany Knowlton
Brittany Knowlton
Numerade Educator
03:25

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. $(0,1) \quad$ b. $(0,2) \quad$ c. $(0,-1)$

Regina Hays
Regina Hays
Numerade Educator
03:25

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. $(0,1) \quad$ b. $(0,4) \quad$ c. $(0,5)$

Regina Hays
Regina Hays
Numerade Educator
04:19

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. $(0,1) \quad$ b. $(0,-2)$ $\quad$ c. $(0,1 / 4) \quad$ d. $(-1,-1)$

Regina Hays
Regina Hays
Numerade Educator
04:10

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. $(0,1) \quad$ b. $(0,2)$ $\quad$ c. $(0,-1) \quad$ d. $(0,0)$

Regina Hays
Regina Hays
Numerade Educator
05:26

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. $(0,-1) \quad$ b. $(0,1)$ $\quad$ c. $(0,3) \quad$ d. $(1,-1)$

Regina Hays
Regina Hays
Numerade Educator
03:38

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. $(0,2) \quad$ b. $(0,-6) \quad$ c. $(-2 \sqrt{3},-4)$

Regina Hays
Regina Hays
Numerade Educator
02:24

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

Nick Johnson
Nick Johnson
Numerade Educator
04:23

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 Hays
Regina Hays
Numerade Educator
01:02

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 Noh
Clarissa Noh
Numerade Educator
07:43

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 Knowlton
Brittany Knowlton
Numerade Educator
03:31

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$x \quad$ b. $\sin 2 x$ $\quad$ c. 3$e^{x / 2} \quad$ d. 2$e^{-x / 2} \cos 2 x$
Graph all four solutions over the interval $-2 \leq x \leq 6$ to compare the results.

Clarissa Noh
Clarissa Noh
Numerade Educator
14:31

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 Hays
Regina Hays
Numerade Educator
04:03

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 Noh
Clarissa Noh
Numerade Educator
06:13

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 Knowlton
Brittany Knowlton
Numerade Educator
03:53

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

Clarissa Noh
Clarissa Noh
Numerade Educator
04:35

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 Knowlton
Brittany Knowlton
Numerade Educator
17:01

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.

$y^{\prime}=x+y, \quad y(0)=-7 / 10 ; \quad-4 \leq x \leq 4, \quad-4 \leq y \leq 4;$
$b=1$

Regina Hays
Regina Hays
Numerade Educator
15:30

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.

$y^{\prime}=-x / y, \quad y(0)=2 ; \quad-3 \leq x \leq 3, \quad-3 \leq y \leq 3;$
$b=2$

Regina Hays
Regina Hays
Numerade Educator
21:30

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.

$y^{\prime}=y(2-y), \quad y(0)=1 / 2 ; \quad 0 \leq x \leq 4,0 \leq y \leq 3;$
$b=3$

Regina Hays
Regina Hays
Numerade Educator
20:44

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.

$y^{\prime}=(\sin x)(\sin y), \quad y(0)=2 ; \quad-6 \leq x \leq 6, \quad-6 \leq y \leq 6;$
$b=3 \pi / 2$

Regina Hays
Regina Hays
Numerade Educator