Chapter 6 Differential Equations And Mathematical Modeling Answers

Problem 1

In Exercises $1-10,$ find the general solution to the exact differential
equation.
$\frac{d y}{d x}=5 x^{4}-\sec ^{2} x$

Amrita B.

Numerade Educator

Problem 2

In Exercises $1-10,$ find the general solution to the exact differential equation.
$$\frac{d y}{d x}=\sec x \tan x-e^{x}$$

Bahar T.

Numerade Educator

Problem 3

In Exercises $1-10,$ find the general solution to the exact differential equation.
$$\frac{d y}{d x}=\sin x-e^{-x}+8 x^{3}$$

Amrita B.

Numerade Educator

Problem 4

In Exercises $1-10,$ find the general solution to the exact differential equation.
$$\frac{d y}{d x}=\frac{1}{x}-\frac{1}{x^{2}}(x>0)$$

Bahar T.

Numerade Educator

Problem 5

In Exercises $1-10,$ find the general solution to the exact differential equation.
$$\frac{d y}{d x}=5^{x} \ln 5+\frac{1}{x^{2}+1}$$

Amrita B.

Numerade Educator

Problem 6

In Exercises $1-10,$ find the general solution to the exact differential equation.
$$\frac{d y}{d x}=\frac{1}{\sqrt{1-x^{2}}}-\frac{1}{\sqrt{x}}$$

Bahar T.

Numerade Educator

Problem 7

In Exercises $1-10,$ find the general solution to the exact differential equation.
$$\frac{d y}{d t}=3 t^{2} \cos \left(t^{3}\right)$$

Amrita B.

Numerade Educator

Problem 8

In Exercises $1-10,$ find the general solution to the exact differential equation.
$$\frac{d y}{d t}=(\cos t) e^{\sin t}$$

Bahar T.

Numerade Educator

Problem 9

In Exercises $1-10,$ find the general solution to the exact differential equation.
$$\frac{d u}{d x}=\left(\sec ^{2} x^{5}\right)\left(5 x^{4}\right)$$

Amrita B.

Numerade Educator

Problem 10

In Exercises $1-10,$ find the general solution to the exact differential equation.
$$\frac{d y}{d u}=4(\sin u)^{3}(\cos u)$$

Bahar T.

Numerade Educator

Problem 11

In Exercises $11-20,$ solve the initial value problem explicitly.
$$\frac{d y}{d x}=3 \sin x$ and $y=2$ when $x=0$$

Amrita B.

Numerade Educator

Problem 12

In Exercises $11-20,$ solve the initial value problem explicitly.
$$\frac{d y}{d x}=2 e^{x}-\cos x$ and $y=3$ when $x=0$$

Bahar T.

Numerade Educator

Problem 13

In Exercises $11-20,$ solve the initial value problem explicitly.
$$\frac{d u}{d x}=7 x^{6}-3 x^{2}+5$ and $u=1$ when $x=1$$

Amrita B.

Numerade Educator

Problem 14

In Exercises $11-20,$ solve the initial value problem explicitly.
$$\frac{d A}{d x}=10 x^{9}+5 x^{4}-2 x+4$ and $A=6$ when $x=1$$

Bahar T.

Numerade Educator

Problem 15

In Exercises $11-20,$ solve the initial value problem explicitly.
$$\frac{d y}{d x}=-\frac{1}{x^{2}}-\frac{3}{x^{4}}+12$ and $y=3$ when $x=1$$

Amrita B.

Numerade Educator

Problem 16

In Exercises $11-20,$ solve the initial value problem explicitly.
$$\frac{d y}{d x}=5 \sec ^{2} x-\frac{3}{2} \sqrt{x}$ and $y=7$ when $x=0$$

Bahar T.

Numerade Educator

Problem 17

In Exercises $11-20,$ solve the initial value problem explicitly.
$\frac{d y}{d t}=\frac{1}{1+t^{2}}+2^{t} \ln 2$ and $y=3$ when $t=0$

Amrita B.

Numerade Educator

Problem 18

In Exercises $11-20,$ solve the initial value problem explicitly.
$\frac{d x}{d t}=\frac{1}{t}-\frac{1}{t^{2}}+6$ and $x=0$ when $t=1$

Bahar T.

Numerade Educator

Problem 19

In Exercises $11-20,$ solve the initial value problem explicitly.
$\frac{d v}{d t}=4 \sec t \tan t+e^{t}+6 t$ and $v=5$ when $t=0$

Amrita B.

Numerade Educator

Problem 20

In Exercises $11-20,$ solve the initial value problem explicitly.
$$y=4 \sec t+e^{t}+3 t^{2}(-\pi / 2$$
$$\frac{d s}{d t}=t(3 t-2)$ and $s=0$ when $t=1$$

Bahar T.

Numerade Educator

Problem 21

In Exercises $21-24$ , solve the initial value problem using the Fundamental Theorem. (Your answer will contain a definite integral.)
$\frac{d y}{d x}=\sin \left(x^{2}\right)$ and $y=5$ when $x=1$

Amrita B.

Numerade Educator

Problem 22

In Exercises $21-24$ , solve the initial value problem using the Fundamental Theorem. (Your answer will contain a definite integral.)
$\frac{d u}{d x}=\sqrt{2+\cos x}$ and $u=-3$ when $x=0$

Bahar T.

Numerade Educator

Problem 23

In Exercises $21-24$ , solve the initial value problem using the Fundamental Theorem. (Your answer will contain a definite integral.)
$F^{\prime}(x)=e^{\cos x}$ and $F(2)=9$

Amrita B.

Numerade Educator

Problem 24

In Exercises $21-24$ , solve the initial value problem using the Fundamental Theorem. (Your answer will contain a definite integral.)
$G^{\prime}(s)=\sqrt[3]{\tan s}$ and $G(0)=4$

Bahar T.

Numerade Educator

Problem 25

In Exercises $25-28$ , match the differential equation with the graph of a family of functions (a) (d) that solve it. Use slope analysis, not your graphing calculator.
$$\frac{d y}{d x}=(\sin x)^{2}$$

Xiaomeng Z.

Numerade Educator

Problem 26

Xiaomeng Z.

Numerade Educator

Problem 26

Bahar T.

Numerade Educator

Problem 27

Amrita B.

Numerade Educator

Problem 28

Bahar T.

Numerade Educator

Problem 29

In Exercises $29-34$ , construct a slope field for the differential equation. In each case, copy the graph at the right and draw tiny segments through the twelve lattice points shown in the graph.
Use slope analysis, not your graphing calculator.

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

Amrita B.

Numerade Educator

Problem 30

Bahar T.

Numerade Educator

Problem 31

Amrita B.

Numerade Educator

Problem 32

Bahar T.

Numerade Educator

Problem 33

Amrita B.

Numerade Educator

Problem 34

Bahar T.

Numerade Educator

Problem 35

In Exercises $35-40,$ match the differential equation with the appropri-
ate slope field. Then use the slope field to sketch the graph of the par-
ticular solution through the highlighted point $(3,2)$ . (All slope fields
are shown in the window $[-6,6]$ by $[-4,4] .$ )

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

Bobby B.

University of North Texas

Problem 36

Bahar T.

Numerade Educator

Problem 37

Amrita B.

Numerade Educator

Problem 38

Bahar T.

Numerade Educator

Problem 39

Amrita B.

Numerade Educator

Problem 40

Bahar T.

Numerade Educator

Problem 41

In Exercises $41-44,$ use Euler's Method with increments of $\Delta x=0.1$ to approximate the value of $y$ when $x=1.3 .$
$\frac{d y}{d x}=x-1$ and $y=2$ when $x=1$

Xiaomeng Z.

Numerade Educator

Problem 42

In Exercises $41-44,$ use Euler's Method with increments of $\Delta x=0.1$ to approximate the value of $y$ when $x=1.3 .$
$\frac{d y}{d x}=y-1$ and $y=3$ when $x=1$

Bahar T.

Numerade Educator

Problem 43

In Exercises $41-44,$ use Euler's Method with increments of $\Delta x=0.1$ to approximate the value of $y$ when $x=1.3 .$
$\frac{d y}{d x}=y-x$ and $y=2$ when $x=1$

Xiaomeng Z.

Numerade Educator

Problem 44

In Exercises $41-44,$ use Euler's Method with increments of $\Delta x=0.1$ to approximate the value of $y$ when $x=1.3 .$
$\frac{d y}{d x}=2 x-y$ and $y=0$ when $x=1$

Bahar T.

Numerade Educator

Problem 45

In Exercises $45-48$ , use Euler's Method with increments of $\Delta x=-0.1$ to approximate the value of $y$ when $x=1.7$
$\frac{d y}{d x}=2-x$ and $y=1$ when $x=2$

Xiaomeng Z.

Numerade Educator

Problem 46

In Exercises $45-48$ , use Euler's Method with increments of $\Delta x=-0.1$ to approximate the value of $y$ when $x=1.7$
$\frac{d y}{d x}=1+y$ and $y=0$ when $x=2$

Bahar T.

Numerade Educator

Problem 47

In Exercises $45-48$ , use Euler's Method with increments of $\Delta x=-0.1$ to approximate the value of $y$ when $x=1.7$
$\frac{d y}{d x}=x-y$ and $y=2$ when $x=2$

Xiaomeng Z.

Numerade Educator

Problem 48

In Exercises $45-48$ , use Euler's Method with increments of $\Delta x=-0.1$ to approximate the value of $y$ when $x=1.7$
$\frac{d y}{d x}=x-2 y$ and $y=1$ when $x=2$

Bahar T.

Numerade Educator

Problem 49

In Exercises 49 and $50,$ (a) determine which graph shows the solution of the initial value problem without actually solving the problem. (b) Writing to Learn Explain how you eliminated two of the
possibilities.
$$\frac{d y}{d x}=\frac{1}{1+x^{2}}, \quad y(0)=\frac{\pi}{2}$$

Xiaomeng Z.

Numerade Educator

Problem 50

Bahar T.

Numerade Educator

Problem 51

Writing to Learn Explain why $y=x^{2}$ could not be a solution
to the differential equation with slope field shown below.

Amrita B.

Numerade Educator

Problem 52

Writing to Learn Explain why $y=\sin x$ could not be a
solution to the differential equation with slope field shown
below.

Bahar T.

Numerade Educator

Problem 53

Percentage Error Let $y=f(x)$ be the solution to the initial
value problem $d y / d x=2 x+1$ such that $f(1)=3 .$ Find the per-
centage error if Euler's Method with $\Delta x=0.1$ is used to approxi-
mate $f(1.4) .$

Amrita B.

Numerade Educator

Problem 54

Percentage Error Let $y=f(x)$ be solution to the initial
value problem $d y / d x=2 x-1$ such that $f(2)=3 .$ Find the per-
centage error if Euler's Method with $\Delta x=-0.1$ is used to ap-
proximate $f(1.6) .$

Xiaomeng Z.

Numerade Educator

Problem 55

Perpendicular Slope Fields The figure below shows the
slope fields for the differential equations $d y / d x=e^{(x-y) / 2}$ and
$d y / d x=-e^{(y-x) / 2}$ superimposed on the same grid. It appears that
the slope lines are perpendicular wherever they intersect. Prove
algebraically that this must be so.

Amrita B.

Numerade Educator

Problem 56

Perpendicular Slope Fields If the slope fields for the differ-
ential equations $d y / d x=\sec x$ and $d y / d x=g(x)$ are perpendicu-
lar (as in Exercise $55 ),$ find $g(x)$ .

Bahar T.

Numerade Educator

Problem 57

Plowing Through a Slope Field The slope field for the dif-
ferential equation $d y / d x=\csc x$ is shown below. Find a function
that will be perpendicular to every line it crosses in the slope
field. (Hint: First find a differential equation that will produce a
perpendicular slope field.)

Amrita B.

Numerade Educator

Problem 58

Plowing Through a Slope Field The slope field for the dif-
ferential equation $d y / d x=1 / x$ is shown below. Find a function
that will be perpendicular to every line it crosses in the slope
field. (Hint: First first find a differential equation that will produce a
perpendicular slope field.)

Xiaomeng Z.

Numerade Educator

Problem 59

True or False Any two solutions to the differential equation
$d v / d x=5$ are parallel lines. Justify your answer.

Amrita B.

Numerade Educator

Problem 60

True or False If $f(x)$ is a solution to $d y / d x=2 x$ , then $f^{-1}(x)$ is
a solution to $d y / d x=2 y .$ Justify your answer.

Bahar T.

Numerade Educator

Problem 61

Multiple Choice A slope field for the differential equation
$d y / d x=42-y$ will show

Amrita B.

Numerade Educator

Problem 61

Multiple Choice A slope field for the differential equation
$d y / d x=42-y$ will show
(A) a line with slope $-1$ and $y$ -intercept 42 .
(B) a vertical asymptote at $x=42$ .
(C) a horizontal asymptote at $y=42$
(D) a family of parabolas opening downward.
(E) a family of parabolas opening to the left.

Xiaomeng Z.

Numerade Educator

Problem 62

Multiple Choice For which of the following differential equa-
tions will a slope field show nothing but negative slopes in the
fourth quadrant?
(A) $\frac{d y}{d x}=-\frac{x}{y} \quad$ (B) $\frac{d y}{d x}-x y+5 \quad$ (C) $\frac{d y}{d x}=x y^{2}-2$
(D) $\frac{d y}{d x}=\frac{x^{3}}{y^{2}} \quad$ (E) $\frac{d y}{d x}=\frac{y}{x^{2}}-3$

Bahar T.

Numerade Educator

Problem 63

Multiple Choice If $d y / d x=2 x y$ and $y=1$ when $x=0,$ then
$y=B$
(A) $y^{2 x}$ (B) $e^{x^{2}}$ (C) $x^{2} y$ (D) $x^{2} y+1 \quad$ (E) $\frac{x^{2} y^{2}}{2}+1$

Amrita B.

Numerade Educator

Problem 64

Multiple Choice Which of the following differential equa-
tions would produce the slope field shown below?

(A) $\frac{d y}{d x}=y-|x| \quad$ (B) $\frac{d y}{d x}=|y|-x$
(C) $\frac{d y}{d x}=|y-x| \quad$ (D) $\frac{d y}{d x}=|y+x|$
(E) $\frac{d y}{d x}=|y|-|x|$

Bahar T.

Numerade Educator

Problem 65

Solving Differential Equations Let $\frac{d y}{d x}=x-\frac{1}{x^{2}}$
(a) Find a solution to the differential equation in the interval
$(0,)$ that satisfies $y(1)=2$
(b) Find a solution to the differential equation in the interval
$(-\infty, 0)$ that satisfies $y(-1)=1$

Xiaomeng Z.

Numerade Educator

Problem 65

Solving Differential Equations Let $\frac{d y}{d x}=x-\frac{1}{x^{2}}$
(a) Find a solution to the differential equation in the interval
$(0,)$ that satisties $y(1)=2$
(b) Find a solution to the differential equation in the interval
$(-\infty, 0)$ that satisfies $y(-1)=1$
(c) Show that the following piecewise function is a solution to
the differential equation for any values of $C_{1}$ and $C_{2}$ .
$y=\left\{\begin{array}{l}{\frac{1}{x}+\frac{x^{2}}{2}+C_{1}} \\ {\frac{1}{x}+\frac{x^{2}}{2}+C_{2}}\end{array}\right.$$x<0$ $x>0$
(d) Choose values for $C_{1}$ and $C_{2}$ so that the solution in
part (c) agrees with the solutions in parts (a) and (b).
(e) Choose values for $C_{1}$ and $C_{2}$ so that the solution in
part (c) satisfies $y(2)=-1$ and $y(-2)=2$

Problem 66

Solving Differential Equations Let $\frac{d y}{d x}=\frac{1}{x}$ .
(a) Show that $y=\ln x+C$ is a solution to the differential
equation in the interval $(0, \infty)$
(b) Show that $y=\ln (-x)+C$ is a solution to the differential
equation in the interval $(-\infty, 0)$
(c) Writing to Learn Explain why $y=\ln |x|+C$ is
a solution to the differential equation in the domain
$(-\infty, 0) \cup(0, \infty)$
(d) Show that the function
$y=\left\{\begin{array}{l}{\ln (-x)+C_{1}} \\ {\ln x+C_{2}}\end{array}\right.$ $x<0$ $x>0$
is a solution to the differential equation for any values of
$C_{1}$ and $C_{2}$

Bahar T.

Numerade Educator

Problem 67

Second-Order Differential Equations Find the general so-
lution to each of the following second-order differential equa-
tions by first finding $d y / d x$ and then finding $y$ . The general solu-
tion will have two unknown constants.
(a) $\frac{d^{2} y}{d x^{2}}=12 x+4$ (b)$\frac{d^{2} y}{d x^{2}}=e^{x}+\sin x$
(c) $\frac{d^{2} y}{d x^{2}}=x^{3}+x^{-3}$

Xiaomeng Z.

Numerade Educator

Problem 68

Second-Order Differential Equations Find the specific solution to each of the following second-order initial value problems by first finding $d y / d x$ and then finding $y$ .
(a) $\frac{d^{2} y}{d x^{2}}=24 x^{2}-10$ when $x=1, \frac{d y}{d x}=3$ and $y=5$
(b) $\frac{d^{2} y}{d x^{2}}=\cos x-\sin x when $x=0, \frac{d y}{d x}=2$ and $y=0$
(c) $\frac{d^{2} y}{d x^{2}}=e^{x}-x$ when $x=0, \frac{d y}{d x}=0$ and $y=1$

Bahar T.

Numerade Educator

Problem 69

Differential Equation Potpourri For each of the following differential equations, find at least one particular solution. You will need to call on past experience with functions you have differentiated. For a greater challenge, find the general solution.
(a) $y^{\prime}=x$ (b)$y^{\prime}=-x$ (c)$y^{\prime}=y$
(d)$y^{\prime}=-y$ (e)$y^{\prime \prime}=-y$

Xiaomeng Z.

Numerade Educator

Problem 70

Second-Order Potpourri For each of the following second-order differential equations, find at least one particular solution. You will need to call on past experience with functions you have differentiated. For a significantly greater challenge, find the general solution (which will involve two unknown constants)
(a)$y^{\prime \prime}=x$ (b)$y^{\prime \prime}=-x$ (c)$y^{\prime \prime}=-\sin x$
(d)$y^{n}=y$ (e)$y^{\prime \prime}=-y$

Bahar T.

Numerade Educator