Match the differential equations with their slope fields, graphed here. (GRAPHS CANNOT COPY)

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

Regina H.

Numerade Educator

Match the differential equations with their slope fields, graphed here. (GRAPHS CANNOT COPY)

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

Regina H.

Numerade Educator

Match the differential equations with their slope fields, graphed here. (GRAPHS CANNOT COPY)

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

Regina H.

Numerade Educator

Match the differential equations with their slope fields, graphed here. (GRAPHS CANNOT COPY)

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

Regina H.

Numerade Educator

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

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

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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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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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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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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Write an equivalent first-order differential equation and initial condition for $y.$

$$y=x+4+\int_{-2}^{x} t e^{y(n)} d t$$

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Write an equivalent first-order differential equation and initial condition for $y.$

$$y=\ln x+\int_{x}^{e} \sqrt{t^{2}+(y(t))^{2}} d t$$

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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)

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(GRAPH CANNOT COPY)

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

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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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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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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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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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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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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 $y(1) ?$

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

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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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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

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

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

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

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

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

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

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

Have no explicit solution in terms of elementary functions. Use a CAS to explore graphically each of the differential equations.

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

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Have no explicit solution in terms of elementary functions. Use a CAS to explore graphically each of the differential equations.

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

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

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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

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

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 $x^{*}.$

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

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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 $x^{*}.$

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

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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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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 $x^{*}.$

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

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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.

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

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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.

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

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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.

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

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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.

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

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