# Thomas Calculus 12

## Educators

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)$$\quad b. (0,2) \quad c. (0,-1) Check back soon! Problem 24 In Exercises 23-28, obtain a slope field and add to it graphs of the solution curves passing through the given points.$$\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$$Check back soon! Problem 31 Exercises 31 and 32 have no explicit solution in terms of elementary functions. Use a CAS to explore graphically each of the differential equations. y^{\prime}=\cos (2 x-y), \quad y(0)=2 0 \leq x \leq 5, \quad 0 \leq y \leq 5 Check back soon! Problem 32 A Gompertz equation y^{\prime}=y(1 / 2-\ln y), \quad y(0)=1 / 3 0 \leq x \leq 4, \quad 0 \leq y \leq 3 Check back soon! Problem 33 Use a CAS to find the solutions of y^{\prime}+y=f(x) subject to the initial condition y(0)=0, if f(x) is a. 2x \quad b. \sin 2 x c. 3e^{x / 2} \quad d. 2e^{-x / 2} \cos 2 x . Graph all four solutions over the interval -2 \leq x \leq 6 to com- pare the results. Check back soon! Problem 34 a. Use a CAS to plot the slope field of the differential equation y^{\prime}=\frac{3 x^{2}+4 x+2}{2(y-1)} b. Separate the variables and use a CAS integrator to find the general solution in implicit form. c. Using a CAS implicit function grapher, plot solution curves for the arbitrary constant values C=-6,-4,-2,0,2,4,6 . d. Find and graph the solution that satisfies the initial condition y(0)=-1 . Check back soon! Problem 35 In Exercises 35-38, use Euler's method with the specified step size to estimate the value of the solution at the given point x^{*} . Find the value of the exact solution at x^{*} .$$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$
$b=3 \pi / 2$\$

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