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Calculus: Graphical, Numerical, Algebraic

Ross L. Finney, Franklin D. Demana, Bet K. Waits,

Chapter 7

Differential Equations and Mathematical Modeling - all with Video Answers

Educators

Section 1

Slope Fields and Euler's Method

01:08

Problem 1

Find the general solution to the exact differential equation.
$\frac{d y}{d x}=5 x^{4}-\sec ^{2} x$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:22

Problem 2

Find the general solution to the exact differential equation.
$\frac{d y}{d x}=\sec x \tan x-e^{x}$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:26

Problem 3

Find the general solution to the exact differential equation.
$\frac{d y}{d x}=\sin x-e^{-x}+8 x^{3}$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:17

Problem 4

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

Charles Machakwa
Charles Machakwa
Numerade Educator
02:27

Problem 5

Find the general solution to the exact differential equation.
$\frac{d y}{d x}=5^{x} \ln 5+\frac{1}{x^{2}+1}$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:46

Problem 6

Find the general solution to the exact differential equation.
$\frac{d y}{d x}=\frac{1}{\sqrt{1-x^{2}}}-\frac{1}{\sqrt{x}}$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:48

Problem 7

Find the general solution to the exact differential equation.
$\frac{d y}{d t}=3 t^{2} \cos \left(t^{3}\right)$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:25

Problem 8

Find the general solution to the exact differential equation.
$\frac{d y}{d t}=(\cos t) e^{\sin t}$

Charles Machakwa
Charles Machakwa
Numerade Educator
03:06

Problem 9

Find the general solution to the exact differential equation.
$\frac{d u}{d x}=\left(\sec ^{2} x^{5}\right)\left(5 x^{4}\right)$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:41

Problem 10

Find the general solution to the exact differential equation.
$\frac{d y}{d u}=4(\sin u)^{3}(\cos u)$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:42

Problem 11

Solve the initial value problem explicitly.
$\frac{d y}{d x}=3 \sin x$ and $y=2$ when $x=0$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:41

Problem 12

Solve the initial value problem explicitly.
$\frac{d y}{d x}=2 e^{x}-\cos x$ and $y=3$ when $x=0$

Charles Machakwa
Charles Machakwa
Numerade Educator
02:01

Problem 13

Solve the initial value problem explicitly.
$\frac{d u}{d x}=7 x^{6}-3 x^{2}+5$ and $u=1$ when $x=1$

Charles Machakwa
Charles Machakwa
Numerade Educator
02:31

Problem 14

Solve the initial value problem explicitly.
$\frac{d A}{d x}=10 x^{9}+5 x^{4}-2 x+4$ and $A=6$ when $x=1$

Charles Machakwa
Charles Machakwa
Numerade Educator
02:54

Problem 15

Solve the initial value problem explicitly.
$\frac{d y}{d x}=-\frac{1}{x^{2}}-\frac{3}{x^{4}}+12$ and $y=3$ when $x=1$

Charles Machakwa
Charles Machakwa
Numerade Educator
02:30

Problem 16

Solve the initial value problem explicitly.
$\frac{d y}{d x}=5 \sec ^{2} x-\frac{3}{2} \sqrt{x}$ and $y=7$ when $x=0$

Charles Machakwa
Charles Machakwa
Numerade Educator
02:26

Problem 17

Solve the initial value problem explicitly.
$\frac{d y}{d t}=\frac{1}{1+t^{2}}+2^{t} \ln 2$ and $y=3$ when $t=0$

Charles Machakwa
Charles Machakwa
Numerade Educator
02:31

Problem 18

Solve the initial value problem explicitly.
$\frac{d x}{d t}=\frac{1}{t}-\frac{1}{t^{2}}+6$ and $x=0$ when $t=1$

Charles Machakwa
Charles Machakwa
Numerade Educator
03:03

Problem 19

Solve the initial value problem explicitly.
$\frac{d v}{d t}=4 \sec t \tan t+e^{t}+6 t$ and $v=5$ when $t=0$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:51

Problem 20

Solve the initial value problem explicitly.
$\frac{d s}{d t}=t(3 t-2)$ and $s=0$ when $t=1$

Charles Machakwa
Charles Machakwa
Numerade Educator
01:39

Problem 21

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 $x=1$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:50

Problem 22

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 $x=0$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:31

Problem 23

Solve the initial value problem using the Fundamental Theorem. (Your answer will contain a definite integral.
$F^{\prime}(x)=e^{\cos x}$ and $F(2)=9$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:37

Problem 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 $G(0)=4$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:27

Problem 25

Match the differential equation with the graph of a family of functions (a)-(d) at the top of the next page that solve it. Use slope analysis, not your graphing calculator.
$\frac{d y}{d x}=(\sin x)^{2}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:23

Problem 26

Match the differential equation with the graph of a family of functions (a)-(d) at the top of the next page that solve it. Use slope analysis, not your graphing calculator.
$\frac{d y}{d x}=(\sin x)^{3}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:21

Problem 27

Match the differential equation with the graph of a family of functions (a)-(d) at the top of the next page that solve it. Use slope analysis, not your graphing calculator.
$\frac{d y}{d x}=(\cos x)^{2}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:43

Problem 28

Match the differential equation with the graph of a family of functions (a)-(d) at the top of the next page that solve it. Use slope analysis, not your graphing calculator.
$\frac{d y}{d x}=(\cos x)^{3}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:12

Problem 29

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:46

Problem 30

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

Madi Sousa
Madi Sousa
Numerade Educator
01:50

Problem 31

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:39

Problem 32

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:27

Problem 33

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:31

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:36

Problem 35

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(3,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=x$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:38

Problem 36

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(3,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=y$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:45

Problem 37

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(3,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=x-y$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:11

Problem 38

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(3,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=y-x$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:51

Problem 39

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(3,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=-\frac{y}{x}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:15

Problem 40

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(3,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=-\frac{x}{y}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:47

Problem 41

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(0,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=\sqrt{x^{2}-x+1}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:38

Problem 42

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(0,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=\sqrt{y^{2}-4 y+5}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:20

Problem 43

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(0,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=|x+y|$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:27

Problem 44

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(0,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=|x-y|$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:33

Problem 45

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(0,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=|x|$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:14

Problem 46

Match the differential equation with the appropriate slope field. Then use the slope field to sketch the graph of the particular solution through the highlighted point $(0,2) .$ (All slope fields are shown in the window [-6,6] by $[-4,4] .)$
$\frac{d y}{d x}=x y$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:09

Problem 47

(a) Sketch a graph of the solution to the initial value problem
$$\frac{d y}{d x}=\sec ^{2} x \text { and } y=1 \text { when } x=\pi$$
(b) Writing to Learn A student solved part (a) and used a graphing calculator to produce the following graph:
$$[-2 \pi, 2 \pi] \text { by }[-4,4]$$
How would you explain to this student why this graph is not the correct answer to part (a)?

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:35

Problem 48

(a) Sketch a graph of the solution to the initial value problem
$$\frac{d y}{d x}=-x^{-2} \text {and } y=1 \text { when } x=1$$
(b) Writing to Learn A student solved part (a) and used a graphing calculator to produce the following graph: How would you explain to this student why this graph is not the correct answer to part (a)?

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:45

Problem 49

Line A single line from the slope field for $\frac{d y}{d x}=2 y+x$ is shown in the second quadrant of one of the following three graphs. Choose the only possible graph and draw a line for the same slope field through the reflected point in the first quadrant. All graphs are shown in the window [-4.7,4.7] by $[-3.1,3.1] .$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:49

Problem 50

A single line from the slope field for $\frac{d y}{d x}=y^{2}-x$ is shown in the first quadrant of one of the following three graphs. Choose the only possible graph and draw a line for the same slope field through the reflected point in the second quadrant. All graphs are shown in the window [-4.7,4.7] by [-3.1,3.1]

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
03:28

Problem 51

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 $x=1$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
03:52

Problem 52

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 $x=1$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:57

Problem 53

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 $x=1$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:57

Problem 54

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 $x=1$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
03:11

Problem 55

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 $x=2$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
03:38

Problem 56

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 $x=2$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
03:19

Problem 57

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 $x=2$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
03:24

Problem 58

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 $x=2$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:09

Problem 59

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:35

Problem 60

(a) determine which graph shows the solution of the initial value problem without actually solving the problem.
(b) Explain how you eliminated two of the possibilities.
$\frac{d y}{d x}=-x, \quad y(-1)=1$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:06

Problem 61

Explain why $y=x^{2}$ could not be a solution to the differential equation with slope field shown below.
$$
[-4.7,4.7] \text { by }[-3.1,3.1]
$$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:25

Problem 62

Explain why $y=\sin x$ could not be a solution to the differential equation with slope field shown below.
$$[-4.7,4.7] \text { by }[-3.1,3.1]$$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
06:33

Problem 63

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 percentage error if Euler's Method with $\Delta x=0.1$ is used to approximate $f(1.4)$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
05:49

Problem 64

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
00:53

Problem 65

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 Bhasin
Amrita Bhasin
Numerade Educator
01:25

Problem 66

If the slope fields for the differential equations $d y / d x=\sec x$ and $d y / d x=g(x)$ are perpendicular (as in Exercise 65$),$ find $g(x)$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:48

Problem 67

The slope field for the differential 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. $]$ $$[-4.7,4.7] \text { by }[-3.1,3.1]$$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:21

Problem 68

The slope field for the differential 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 find a differential equation that will produce a perpendicular slope field.]
$$[-4.7,4.7] \text { by }[-3.1,3.1]$$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:14

Problem 69

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:31

Problem 70

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 Tehranipoor
Bahar Tehranipoor
Numerade Educator
01:12

Problem 71

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.

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
03:14

Problem 72

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:22

Problem 73

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:20

Problem 74

Which of the following differential equations would produce the slope field shown below?
$$[-3,3] \text { by }[-1.98,1.98]$$
(A) $\frac{d y}{d x}=y-|x|$
(B) $\frac{d y}{d x}=|y|-x$
(C) $\frac{d y}{d x}=|y-x|$
(D) $\frac{d y}{d x}=|y+x|$
(E) $\frac{d y}{d x}=|y|-|x|$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:44

Problem 75

Let $\frac{d y}{d x}=x-\frac{1}{x^{2}}$
(a) Find a solution to the differential equation in the interval $(0, \infty)$ that satisfies $y(1)=2$
(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 $C_{2}$.
$$y=\left\{\begin{array}{ll}\frac{1}{x}+\frac{x^{2}}{2}+C_{1} & x<0 \\\frac{1}{x}+\frac{x^{2}}{2}+C_{2} & x>0\end{array}\right.$$ (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 $y(-2)=2$

Xiaomeng Zhang
Xiaomeng Zhang
Numerade Educator
04:27

Problem 76

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
$$y=\left\{\begin{array}{ll}\ln (-x)+C_{1}, & x<0 \\\ln x+C_{2}, & x>0\end{array}\right.$$
is a solution to the differential equation for any values of $C_{1}$ and $C_{2}$

Bahar Tehranipoor
Bahar Tehranipoor
Numerade Educator
03:02

Problem 77

Find the general solution to each of the following second-order differential equations by first finding $d y / d x$ and then finding $y$. The general solution 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) $\frac{d^{2} y}{d x^{2}}=x^{3}+x^{-3}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
06:02

Problem 78

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 $y=1$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:52

Problem 79

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:52

Problem 80

For each of the following secondorder 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) $y^{\prime \prime}=y$
(e) $y^{\prime \prime}=-y$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator