Books(current) Courses (current) Earn 💰 Log in(current)

Thomas Calculus 12

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 9

First-Order Differential Equations

Educators


Problem 1

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

Check back soon!

Problem 2

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

Check back soon!

Problem 3

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

Check back soon!

Problem 4

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

Check back soon!

Problem 5

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

Check back soon!

Problem 6

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

Check back soon!

Problem 7

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

Check back soon!

Problem 8

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

Check back soon!

Problem 9

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

Check back soon!

Problem 10

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

Check back soon!

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

Check back soon!

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

Check back soon!

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

Check back soon!

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

Check back soon!

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

Check back soon!

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

Check back soon!

Problem 17

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

Check back soon!

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 $y(2) ?$

Check back soon!

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 $y(5) ?$

Check back soon!

Problem 20

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 $y(2) ?$

Check back soon!

Problem 21

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

Check back soon!

Problem 22

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

Check back soon!

Problem 23

In Exercises $23-28,$ 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. $(0,-1)$

Check back soon!

Problem 24

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

Check back soon!

Problem 25

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

Check back soon!

Problem 26

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

Check back soon!

Problem 27

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

Check back soon!

Problem 28

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

Check back soon!

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

Check back soon!

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

Check back soon!

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.
$y^{\prime}=\cos (2 x-y), \quad y(0)=2$ $0 \leq x \leq 5, \quad 0 \leq y \leq 5$

Check back soon!

Problem 32

A Gompertz equation y^{\prime}=y(1 / 2-\ln y), \quad y(0)=1 / 3
$0 \leq x \leq 4, \quad 0 \leq y \leq 3$

Check back soon!

Problem 33

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. $\sin 2 x$ 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 com-
pare the results.

Check back soon!

Problem 34

a. Use a CAS to plot the slope field of the differential equation
$y^{\prime}=\frac{3 x^{2}+4 x+2}{2(y-1)}$
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
$y(0)=-1 .$

Check back soon!

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

Check back soon!

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 $x^{*}$ .
$$y^{\prime}=2 y^{2}(x-1), \quad y(2)=-1 / 2, \quad d x=0.1, \quad x^{4}=3$$

Check back soon!

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

Check back soon!

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

Check back soon!

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 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 pro-
duced 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.
$$\begin{array}{l}{y^{\prime}=x+y, \quad y(0)=-7 / 10 ; \quad-4 \leq x \leq 4, \quad-4 \leq y \leq 4} \\ {b=1}\end{array}$$

Check back soon!

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

Check back soon!

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

Check back soon!

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

Check back soon!