Problem 1

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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Problem 2

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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Problem 3

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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Problem 4

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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Problem 5

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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Problem 6

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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Problem 7

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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Problem 8

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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Problem 9

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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Problem 10

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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Problem 11

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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Problem 12

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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Problem 13

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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Problem 14

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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Problem 15

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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Problem 16

$y=3 \sin 2 x$

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Problem 17

$y=e^{-2 x}$

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Problem 18

$y=5 \ln x$

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Problem 19

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

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Problem 20

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

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Problem 21

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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Problem 22

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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Problem 23

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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Problem 24

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

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Problem 25

$y=\sin x$

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Problem 26

$y=\cos x$

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Problem 27

$y=\ln x$

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Problem 28

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

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Problem 29

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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Problem 30

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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Problem 31

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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Problem 32

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

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Problem 33

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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Problem 34

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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Problem 35

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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Problem 36

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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Problem 37

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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Problem 38

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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Problem 39

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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Problem 40

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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Problem 41

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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Problem 42

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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Problem 43

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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Problem 44

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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Problem 45

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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Problem 46

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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Problem 47

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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Problem 48

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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Problem 49

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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Problem 50

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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Problem 51

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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Problem 52

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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Problem 53

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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Problem 54

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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Problem 55

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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Problem 56

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

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

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Problem 57

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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Problem 58

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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Problem 59

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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Problem 60

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

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Problem 61

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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Problem 62

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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Problem 63

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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Problem 64

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

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Problem 65

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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Problem 66

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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Problem 67

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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Problem 68

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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Problem 69

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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Problem 70

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

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Problem 71

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

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Problem 72

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

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Problem 73

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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Problem 74

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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Problem 75

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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Problem 76

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

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Problem 77

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

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Problem 78

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

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Problem 79

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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Problem 80

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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Problem 81

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.

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

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Problem 82

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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Problem 83

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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Problem 84

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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Problem 85

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

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Problem 87

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

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Problem 88

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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Problem 89

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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Problem 90

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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Problem 91

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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Problem 92

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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Problem 93

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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Problem 94

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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Problem 95

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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Problem 96

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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Problem 97

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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Problem 98

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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Problem 99

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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