Problem 1

In Exercises $1-4,$ match the differential equations with their slope fields, graphed here.

$$y^{\prime}=x+y$$

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Problem 2

In Exercises $1-4,$ match the differential equations with their slope fields, graphed here.

$$y^{\prime}=y+1$$

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Problem 3

In Exercises $1-4,$ match the differential equations with their slope fields, graphed here.

$$y^{\prime}=\frac{x}{y}$$

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Problem 4

In Exercises $1-4,$ match the differential equations with their slope fields, graphed here.

$$y^{\prime}=y^{2}-x^{2}$$

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Problem 5

In Exercises 5 and $6,$ copy the slope fields and sketch in some of the solution curves.

$$y^{\prime}=(y+2)(y-2)$$

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Problem 6

In Exercises 5 and $6,$ copy the slope fields and sketch in some of the solution curves.

$$y^{\prime}=y(y+1)(y-1)$$

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Problem 7

In Exercises $7-10$ , write an equivalent first-order differential equation and initial condition for $y .$

$$y=-1+\int_{1}^{x}(t-y(t)) d t$$

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Problem 8

In Exercises $7-10$ , write an equivalent first-order differential equation and initial condition for $y .$

$$y=\int_{1}^{x} \frac{1}{t} d t$$

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Problem 9

In Exercises $7-10$ , write an equivalent first-order differential equation and initial condition for $y .$

$$y=2-\int_{0}^{x}(1+y(t)) \sin t d t$$

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Problem 10

In Exercises $7-10$ , write an equivalent first-order differential equation and initial condition for $y .$

$$y=1+\int_{0}^{x} y(t) d t$$

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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

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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 $y(1) ?$

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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 $y(2) ?$

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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 $y(5) ?$

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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 $y(2) ?$

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Problem 21

Show that the solution of the initial value problem

$y^{\prime}=x+y, \quad y\left(x_{0}\right)=y_{0}$

$y=-1-x+\left(1+x_{0}+y_{0}\right) e^{x-x_{0}}$

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Problem 22

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

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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. $(0,-1)$

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Problem 24

In Exercises $23-28,$ obtain a slope field and add to it graphs of the solution curves passing through the given points.

$$\begin{array}{l}{y^{\prime}=2(y-4) \text { with }} \\ {\text { a. }(0,1) \quad \text { b. }(0,4) \quad \text { c. }(0,5)}\end{array}$$

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Problem 25

In Exercises $23-28,$ obtain a slope field and add to it graphs of the solution curves passing through the given points.

$$\begin{array}{l}{y^{\prime}=y(x+y) \text { with }} \\ {\text { a. }(0,1) \quad \text { b. }(0,-2) \quad \text { c. }(0,1 / 4) \quad \text { d. }(-1,-1)}\end{array}$$

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Problem 26

In Exercises $23-28,$ obtain a slope field and add to it graphs of the solution curves passing through the given points.

$$y^{\prime}=y^{2}$$

$$\begin{array}{llll}{\text { a. }(0,1)} & {\text { b. }(0,2)} & {\text { c. }(0,-1)} & {\text { d. }(0,0)}\end{array}$$

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Problem 27

In Exercises $23-28,$ obtain a slope field and add to it graphs of the solution curves passing through the given points.

$$y^{\prime}=(y-1)(x+2)$$

$$\begin{array}{llll}{\text { a. }(0,-1)} & {\text { b. }(0,1)} & {\text { c. }(0,3)} & {\text { d. }(1,-1)}\end{array}$$

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Problem 28

In Exercises $23-28,$ obtain a slope field and add to it graphs of the solution curves passing through the given points.

$$\begin{array}{l}{y^{\prime}=\frac{x y}{x^{2}+4} \text { with }} \\ {\text { a. }(0,2) \quad \text { b. }(0,-6) \quad \text { c. }(-2 \sqrt{3},-4)}\end{array}$$

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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

$$0 \leq x \leq 4, \quad 0 \leq y \leq 3$$

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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.

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

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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.

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

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Problem 32

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

$0 \leq x \leq 4, \quad 0 \leq y \leq 3$

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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.

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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

$y(0)=-1 .$

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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 $x^{*}$ .

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

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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 $x^{*}$ .

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

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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 $x^{*}$ .

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

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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 $x^{*}$ .

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

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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.

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

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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.

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

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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$$

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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$$

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