In Exercises $1-8,$ verify the solution of the differential equation.

$$\begin{array}{ll}{\frac{\text {Solution}}{y=C e^{4 x}}} & {\frac{\text { Differential Equation }}{y^{\prime}=4 y}}\end{array}$$

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In Exercises $1-8,$ verify the solution of the differential equation.

$$\begin{array}{ll}{\frac{\text {Solution}}{y=e^{-2 x}}} & {\frac{\text { Differential Equation }}{y^{\prime}=4 y}}\end{array}$$

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In Exercises $1-8,$ verify the solution of the differential equation.

$$\begin{array}{ll}{\frac{\text {Solution}}{x^{2}+y^{2}=C y}} & {\frac{\text { Differential Equation }}{y^{\prime}=2 x y /\left(x^{2}-y^{2}\right)}}\end{array}$$

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In Exercises $1-8,$ verify the solution of the differential equation.

$$\begin{array}{ll}{\frac{\text {Solution}}{y^{2}-2 \ln y=x^{2}}} & {\frac{\text { Differential Equation }}{\frac{d y}{d x}=\frac{x y}{y^{2}-1}}}\end{array}$$

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In Exercises $1-8,$ verify the solution of the differential equation.

$$\begin{array}{ll}{\frac{\text {Solution}}{y=C_{1} \sin x-C_{2} \cos x}} & {\frac{\text { Differential Equation }}{y^{\prime \prime}+y=0}}\end{array}$$

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In Exercises $1-8,$ verify the solution of the differential equation.

$$\begin{array}{ll}{\frac{\text {Solution}}{y=C_{1} e^{-x} \cos x+C_{2} e^{-x} \sin x}} & {\frac{\text { Differential Equation }}{y^{\prime \prime}+2 y^{\prime}+2 y=0}}\end{array}$$

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In Exercises $1-8,$ verify the solution of the differential equation.

$$\begin{array}{ll}{\frac{\text {Solution}}{y=-\cos x \ln |\sec x+\tan x|}} & {\frac{\text { Differential Equation }}{y^{\prime \prime}+y=\tan x}}\end{array}$$

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In Exercises $1-8,$ verify the solution of the differential equation.

$$\begin{array}{ll}{\frac{\text {Solution}}{y=\frac{2}{5}\left(e^{-4 x}+e^{x}\right)}} & {\frac{\text { Differential Equation }}{y^{\prime \prime}+4 y^{\prime}=2 e^{x}}}\end{array}$$

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In Exercises $9-12$ , verify the particular solution of the differential equation.

$\begin{array}{cc}{\text { Differential Equation }} \\ {\text { Solution }} & {\text { and Initial Condition }} \\ {y=\sin x \cos x-\cos ^{2} x} & {2 y+y^{\prime}=2 \sin (2 x)-1} \\ {} & {y\left(\frac{\pi}{4}\right)=0}\end{array}$

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In Exercises $9-12$ , verify the particular solution of the differential equation.

$\begin{array}{cc}{\text { Differential Equation }} \\ {\text { Solution }} & {\text { and Initial Condition }} \\ {y=\frac{1}{2} x^{2}-2 \cos x-3} & {y^{\prime}=x+2 \sin x} \\ {} & {y(0)=-5}\end{array}$

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In Exercises $9-12$ , verify the particular solution of the differential equation.

$\begin{array}{cc}{\text { Differential Equation }} \\ {\text { Solution }} & {\text { and Initial Condition }} \\ {y=4 e^{-6 x^{2}}} & {y^{\prime}=-12 x y} \\ {} & {y(0)=4}\end{array}$

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In Exercises $9-12$ , verify the particular solution of the differential equation.

$\begin{array}{cc}{\text { Differential Equation }} \\ {\text { Solution }} & {\text { and Initial Condition }} \\ {y=e^{-\cos x}} & {y^{\prime}=y \sin x} \\ {} & {y\left(\frac{\pi}{2}\right)=1}\end{array}$

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In Exercises $13-20,$ determine whether the function is a solution of the differential equation $y^{(4)}-16 y=0$

$y=3 \cos x$

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In Exercises $13-20,$ determine whether the function is a solution of the differential equation $y^{(4)}-16 y=0$

$y=2 \sin x$

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In Exercises $13-20,$ determine whether the function is a solution of the differential equation $y^{(4)}-16 y=0$

$y=3 \cos 2 x$

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$y=3 \sin 2 x$

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$y=e^{-2 x}$

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$y=5 \ln x$

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$y=C_{1} e^{2 x}+C_{2} e^{-2 x}+C_{3} \sin 2 x+C_{4} \cos 2 x$

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$y=3 e^{2 x}-4 \sin 2 x$

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In Exercises $21-28$ , determine whether the function is a solution of the differential equation $x y^{\prime}-2 y=x^{3} e^{x} .$

$y=x^{2}$

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In Exercises $21-28$ , determine whether the function is a solution of the differential equation $x y^{\prime}-2 y=x^{3} e^{x} .$

$y=x^{3}$

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In Exercises $21-28$ , determine whether the function is a solution of the differential equation $x y^{\prime}-2 y=x^{3} e^{x} .$

$y=x^{2} e^{x}$

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$y=x^{2}\left(2+e^{x}\right)$

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$y=\sin x$

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$y=\cos x$

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$y=\ln x$

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$y=x^{2} e^{x}-5 x^{2}$

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In Exercises $29-32$ , some of the curves corresponding to different values of $C$ in the general solution of the differential equation are given. Find the particular solution that passes through the point shown on the graph.

$$\frac{\text { Solution }}{y=C e^{-x / 2}} \quad \frac{\text { Differential Equation }}{2 y^{\prime}+y=0}$$

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In Exercises $29-32$ , some of the curves corresponding to different values of $C$ in the general solution of the differential equation are given. Find the particular solution that passes through the point shown on the graph.

$$\frac{\text { Solution }}{y\left(x^{2}+y\right)=C} \quad \frac{\text { Differential Equation }}{2 x y+\left(x^{2}+2 y\right) y^{\prime}=0}$$

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In Exercises $29-32$ , some of the curves corresponding to different values of $C$ in the general solution of the differential equation are given. Find the particular solution that passes through the point shown on the graph.

$$\frac{\text { Solution }}{y^{2}=C x^{3}} \quad \frac{\text { Differential Equation }}{2 x y^{\prime}-3 y=0}$$

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$$\frac{\text { Solution }}{2 x^{2}-y^{2}=C} \quad \frac{\text { Differential Equation }}{y y^{\prime}-2 x=0}$$

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In Exercises 33 and $34,$ the general solution of the differential equation is given. Use a graphing utility to graph the particular solutions for the given values of $C .$

$4 y y^{\prime}-x=0$

$4 y^{2}-x^{2}=C$

$C=0, C=\pm 1, C=\pm 4$

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In Exercises 33 and $34,$ the general solution of the differential equation is given. Use a graphing utility to graph the particular solutions for the given values of $C .$

$y y^{\prime}+x=0$

$x^{2}+y^{2}=C$

$C=0, C=1, C=4$

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In Exercises $35-40$ , verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

$y=C e^{-2 x}$

$y^{\prime}+2 y=0$

$y=3$ when $x=0$

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In Exercises $35-40$ , verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

$3 x^{2}+2 y^{2}=C$

$3 x+2 y y^{\prime}=0$

$y=3$ when $x=1$

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In Exercises $35-40$ , verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

$y=C_{1} \sin 3 x+C_{2} \cos 3 x$

$y^{\prime \prime}+9 y=0$

$y=2$ when $x=\pi / 6$

$y^{\prime}=1$ when $x=\pi / 6$

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In Exercises $35-40$ , verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

$y=C_{1}+C_{2} \ln x$

$x y^{\prime \prime}+y^{\prime}=0$

$y=0$ when $x=2$

$y^{\prime}=\frac{1}{2}$ when $x=2$

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In Exercises $35-40$ , verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

$y=C_{1} x+C_{2} x^{3}$

$x^{2} y^{\prime \prime}-3 x y^{\prime}+3 y=0$

$y=0$ when $x=2$

$y^{\prime}=4$ when $x=2$

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In Exercises $35-40$ , verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

$y=e^{2 x / 3}\left(C_{1}+C_{2} x\right)$

$9 y^{\prime \prime}-12 y^{\prime}+4 y=0$

$y=4$ when $x=0$

$y=0$ when $x=3$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=6 x^{2}$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=2 x^{3}-3 x$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=\frac{x}{1+x^{2}}$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=\frac{e^{x}}{4+e^{x}}$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=\frac{x-2}{x}$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=x \cos x^{2}$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=\sin 2 x$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=\tan ^{2} x$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=x \sqrt{x-6}$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=2 x \sqrt{3-x}$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=x e^{x^{2}}$

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In Exercises $41-52,$ use integration to find a general solution of the differential equation.

$\frac{d y}{d x}=5 e^{-x / 2}$

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In Exercises $53-56,$ a differential equation and its slope field are given. Complete the table by determining the slopes (if possible) in the slope field at the given points.

$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {-4} & {-2} & {0} & {2} & {4} & {8} \\ \hline y & {2} & {0} & {4} & {4} & {6} & {8} \\ \hline d y / d x & {} & {} & {} \\ \hline\end{array}$$

$\frac{d y}{d x}=\frac{2 x}{y}$

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In Exercises $53-56,$ a differential equation and its slope field are given. Complete the table by determining the slopes (if possible) in the slope field at the given points.

$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {-4} & {-2} & {0} & {2} & {4} & {8} \\ \hline y & {2} & {0} & {4} & {4} & {6} & {8} \\ \hline d y / d x & {} & {} & {} \\ \hline\end{array}$$

$\frac{d y}{d x}=y-x$

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In Exercises $53-56,$ a differential equation and its slope field are given. Complete the table by determining the slopes (if possible) in the slope field at the given points.

$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {-4} & {-2} & {0} & {2} & {4} & {8} \\ \hline y & {2} & {0} & {4} & {4} & {6} & {8} \\ \hline d y / d x & {} & {} & {} \\ \hline\end{array}$$

$\frac{d y}{d x}=x \cos \frac{\pi y}{8}$

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$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {-4} & {-2} & {0} & {2} & {4} & {8} \\ \hline y & {2} & {0} & {4} & {4} & {6} & {8} \\ \hline d y / d x & {} & {} & {} \\ \hline\end{array}$$

$\frac{d y}{d x}=\tan \left(\frac{\pi y}{6}\right)$

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In Exercises $57-60$ , match the differential equation with its slope field. [The slope fields are labeled (a), (b), (c), and (d).]

$\frac{d y}{d x}=\sin (2 x)$

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In Exercises $57-60$ , match the differential equation with its slope field. [The slope fields are labeled (a), (b), (c), and (d).]

$\frac{d y}{d x}=\frac{1}{2} \cos x$

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In Exercises $57-60$ , match the differential equation with its slope field. [The slope fields are labeled (a), (b), (c), and (d).]

$\frac{d y}{d x}=e^{-2 x}$

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$\frac{d y}{d x}=\frac{1}{x}$

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In Exercises $61-64,$ (a) sketch the slope field for the differential equation, (b) use the slope field to sketch the solution that passes through the given point, and (c) discuss the graph of the solution as $x \rightarrow \infty$ and $x \rightarrow-\infty$ . Use a graphing utility to verify your results.

$y^{\prime}=3-x$

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In Exercises $61-64,$ (a) sketch the slope field for the differential equation, (b) use the slope field to sketch the solution that passes through the given point, and (c) discuss the graph of the solution as $x \rightarrow \infty$ and $x \rightarrow-\infty$ . Use a graphing utility to verify your results.

$y^{\prime}=\frac{1}{3} x^{2}-\frac{1}{2} x, \quad(1,1)$

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In Exercises $61-64,$ (a) sketch the slope field for the differential equation, (b) use the slope field to sketch the solution that passes through the given point, and (c) discuss the graph of the solution as $x \rightarrow \infty$ and $x \rightarrow-\infty$ . Use a graphing utility to verify your results.

$y^{\prime}=y-4 x, \quad(2,2)$

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$y^{\prime}=y+x y, \quad(0,-4)$

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Use the slope field for the differential equation $y^{\prime}=1 / x,$ where $x>0,$ to sketch the graph of the solution that satisfies each given initial condition. Then make a conjecture

about the behavior of a particular solution of $y^{\prime}=1 / x$ as $x \rightarrow \infty$ . To print an enlarged copy of the graph, go to the website www. mathgraphs.com.

(a) $(1,0)$

(b) $(2,-1)$

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Use the slope field for the differential equation $y^{\prime}=1 / y,$ where $y>0$ , to sketch the graph of the solution that satisfies each initial condition. Then make a conjecture about the behavior of a particular solution of $y^{\prime}=1 / y$ as $x \rightarrow \infty$ . To print an enlarged copy of the graph, go to the website www. mathgraphs.com.

(a) $(0,1)$

(b) $(1,1)$

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In Exercises $67-72,$ use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition.

$\frac{d y}{d x}=0.25 y, \quad y(0)=4$

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In Exercises $67-72,$ use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition.

$\frac{d y}{d x}=4-y, \quad y(0)=6$

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In Exercises $67-72,$ use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition.

$\frac{d y}{d x}=0.02 y(10-y), \quad y(0)=2$

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$\frac{d y}{d x}=0.2 x(2-y), \quad y(0)=9$

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$\frac{d y}{d x}=0.4 y(3-x), \quad y(0)=1$

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$\frac{d y}{d x}=\frac{1}{2} e^{-x / 8} \sin \frac{\pi y}{4}, \quad y(0)=2$

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In Exercises $73-78$ , use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use $n$ steps of size $h .$

$y^{\prime}=x+y, \quad y(0)=2, \quad n=10, \quad h=0.1$

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In Exercises $73-78$ , use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use $n$ steps of size $h .$

$y^{\prime}=x+y, \quad y(0)=2, \quad n=20, \quad h=0.05$

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In Exercises $73-78$ , use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use $n$ steps of size $h .$

$y^{\prime}=3 x-2 y, \quad y(0)=3, \quad n=10, \quad h=0.05$

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$y^{\prime}=0.5 x(3-y), \quad y(0)=1, \quad n=5, \quad h=0.4$

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$y^{\prime}=e^{x y}, \quad y(0)=1, \quad n=10, \quad h=0.1$

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$y^{\prime}=\cos x+\sin y, \quad y(0)=5, \quad n=10, \quad h=0.1$

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In Exercises $79-81$ , complete the table using the exact solution of the differential equation and two approximations obtained using Euler's Method to approximate the particular solution of

the differential equation. Use $h=0.2$ and $h=0.1$ and compute each approximation to four decimal places.

$$\begin{array}{ll}{\text { Differential }} & {\text { Initial }} & {\text { Exact}}\\ {\text { Equation }} & {\text { Condition }} & {\text { Solution}}\\ {\frac{d y}{d x}=y} & {(0,3)} & {y=3 e^{x}}\end{array}$$

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In Exercises $79-81$ , complete the table using the exact solution of the differential equation and two approximations obtained using Euler's Method to approximate the particular solution of

the differential equation. Use $h=0.2$ and $h=0.1$ and compute each approximation to four decimal places.

$$\begin{array}{ll}{\text { Differential }} & {\text { Initial }} & {\text { Exact}}\\ {\text { Equation }} & {\text { Condition }} & {\text { Solution}}\\ \frac{d y}{d x}=\frac{2 x}{y} & {(0,2)} & {y=\sqrt{2 x^{2}+4}}\end{array}$$

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In Exercises $79-81$ , complete the table using the exact solution of the differential equation and two approximations obtained using Euler's Method to approximate the particular solution of

the differential equation. Use $h=0.2$ and $h=0.1$ and compute each approximation to four decimal places.

$$\begin{array}{ll}{\text { Differential }} & {\text { Initial }} & {\text { Exact}}\\ {\text { Equation }} & {\text { Condition }} & {\text { Solution}}\\ \frac{d y}{d x}=y+\cos (x) & {(0,0)} & {y=\frac{1}{2}\left(\sin x-\cos x+e^{x}\right)}\end{array}$$

Helena R.

Numerade Educator

Compare the values of the approximations in Exercises $79-81$ with the values given by the exact solution. How does the error change as $h$ increases?

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Temperature At time $t=0$ minutes, the temperature of an object is $140^{\circ} \mathrm{F}$ . The temperature of the object is changing at the rate given by the differential equation

$$\frac{d y}{d t}=-\frac{1}{2}(y-72)$$

(a) Use a graphing utility and Euler's Method to approximate the particular solutions of this differential equation at $t=1,2,$ and $3 .$ Use a step size of $h=0.1 .$ (A graphing utility program for Euler's Method is available at the website college. hmco.com.)

(b) Compare your results with the exact solution

$$y=72+68 e^{-t / 2}$$

(c) Repeat parts (a) and (b) using a step size of $h=0.05$ . Compare the results.

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The graph shows a solution of one of the following differential equations. Determine the correct equation. Explain your reasoning.

(a) $y^{\prime}=x y$

(b) $y^{\prime}=\frac{4 x}{y}$

(c) $y^{\prime}=-4 x y$

(d) $y^{\prime}=4-x y$

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In your own words, describe the difference between a general solution of a differential equation and a particular solution.

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Describe how to use Euler's Method to approximate a particular solution of a differential equation.

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It is known that $y=C e^{k x}$ is a solution of the differential equation $y^{\prime}=0.07 y$ . Is it possible to determine $C$ or $k$ from the information given? If so, find its value.

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True or False? In Exercises $89-92$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $y=f(x)$ is a solution of a first-order differential equation, then $y=f(x)+C$ is also a solution.

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True or False? In Exercises $89-92$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The general solution of a differential equation is $y=-4.9 x^{2}+C_{1} x+C_{2} .$ To find a particular solution, you must be given two initial conditions.

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True or False? In Exercises $89-92$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

Slope fields represent the general solutions of differential equations.

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A slope field shows that the slope at the point $(1,1)$ is $6 .$ This slope field represents the family of solutions for the differential equation $y^{\prime}=4 x+2 y$

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Errors and Euler's Method The exact solution of the differential equation

$\frac{d y}{d x}=-2 y$

where $y(0)=4,$ is $y=4 e^{-2 x}$

(a) Use a graphing utility to complete the table, where $y$ is the exact value of the solution, $y_{1}$ is the approximate solution using Euler's Method with $h=0.1, y_{2}$ is the approximate

solution using Euler's Method with $h=0.2, e_{1}$ is the absolute error $\left|y-y_{1}\right|, e_{2}$ is the absolute error $\left|y-y_{2}\right|$ and $r$ is the ratio $e_{1} / e_{2} .$

(b) What can you conclude about the ratio $r$ as $h$ changes?

(c) Predict the absolute error when $h=0.05$

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Errors and Euler's Method Repeat Exercise 93 for which the exact solution of the differential equation

$\frac{d y}{d x}=x-y$

where $y(0)=1,$ is $y=x-1+2 e^{-x}$

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Electric Circuits The diagram shows a simple electric circuit consisting of a power source, a resistor, and an inductor.

A model of the current $I,$ in amperes $(\mathrm{A}),$ at time $t$ is given by the first-order differential equation

$L \frac{d I}{d t}+R I=E(t)$

where $E(t)$ is the voltage $(\mathrm{V})$ produced by the power source, $R$ is the resistance, in ohms $(\Omega),$ and $L$ is the inductance, in henrys (H). Suppose the electric circuit consists of a $24-\mathrm{V}$ power source, a $12-\Omega$ resistor, and a $4-\mathrm{H}$ inductor.

(a) Sketch a slope field for the differential equation.

(b) What is the limiting value of the current? Explain.

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Think About It is known that $y=e^{k t}$ is a solution of the differential equation $y^{\prime \prime}-16 y=0 .$ Find the values of $k .$

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Think About It is known that $y=A \sin \omega t$ is a solution of the differential equation $y^{\prime \prime}+16 y=0 .$ Find the values of $\omega .$

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Let $f$ be a twice-differentiable real-valued function satisfying

$f(x)+f^{\prime \prime}(x)=-x g(x) f^{\prime}(x)$

where $g(x) \geq 0$ for all real $x$ . Prove that $|f(x)|$ is bounded.

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Prove that if the family of integral curves of the differential equation

$\frac{d y}{d x}+p(x) y=q(x), \quad p(x) \cdot q(x) \neq 0$

is cut by the line $x=k,$ the tangents at the points of intersection are concurrent.

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