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Calculus, Early Transcendentals

Dennis G. Zill, Warren S. Wright

Chapter 8

First-Order Differential Equations - all with Video Answers

Educators

Section 1

Separable Equations

00:37

Problem 1

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=\sin 5 x$

Edward Downes
Edward Downes
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01:10

Problem 2

Solve the given differential equation by separation of variables.
$\frac{d y}{d t}=(t+1)^{2}$

Khushbu Rani
Khushbu Rani
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01:36

Problem 3

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=\frac{y^{3}}{x^{2}}$

Darshan Maheshwari
Darshan Maheshwari
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00:46

Problem 4

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=\frac{1}{5 y^{4}}$

Darshan Maheshwari
Darshan Maheshwari
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01:39

Problem 5

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=\left(\frac{1+x}{1+y}\right)^{2}$

Darshan Maheshwari
Darshan Maheshwari
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01:27

Problem 6

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=\sqrt{x y}$

Darshan Maheshwari
Darshan Maheshwari
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01:55

Problem 7

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=\frac{1+5 x^{2}}{x^{2} \sin y}$

Khushbu Rani
Khushbu Rani
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00:55

Problem 8

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=y^{3} \cos x$

Darshan Maheshwari
Darshan Maheshwari
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00:58

Problem 9

Solve the given differential equation by separation of variables.
$x \frac{d y}{d x}=4 y$

Edward Downes
Edward Downes
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01:05

Problem 10

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}+2 x y=0$

Darshan Maheshwari
Darshan Maheshwari
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00:56

Problem 11

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=e^{3 x+2 y}$

Edward Downes
Edward Downes
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02:12

Problem 12

Solve the given differential equation by separation of variables.
$e^{x} y \frac{d y}{d x}=e^{-y}+e^{-2 x-y}$

Minh Le
Minh Le
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Problem 13

Solve the given differential equation by separation of variables.
$\left(\frac{y+1}{x}\right)^{2} \frac{d y}{d x}=y \ln x$

Semhal Abebe
Semhal Abebe
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03:00

Problem 14

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=\left(\frac{2 y+3}{4 x+5}\right)^{2}$

Semhal Abebe
Semhal Abebe
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03:13

Problem 15

Solve the given differential equation by separation of variables.
$\frac{d N}{d t}+N=N t e^{t+2}$

Darshan Maheshwari
Darshan Maheshwari
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01:19

Problem 16

Solve the given differential equation by separation of variables.
$\frac{d Q}{d t}=k(Q-70)$

Edward Downes
Edward Downes
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02:13

Problem 17

Solve the given differential equation by separation of variables.
$\frac{d P}{d t}=5 P-P^{2}$

Darshan Maheshwari
Darshan Maheshwari
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02:51

Problem 18

Solve the given differential equation by separation of variables.
$\frac{d X}{d t}=(10-X)(50-X)$

Darshan Maheshwari
Darshan Maheshwari
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03:48

Problem 19

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=\frac{x y+3 x-y-3}{x y-2 x+4 y-8}$

Darshan Maheshwari
Darshan Maheshwari
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03:19

Problem 20

Solve the given differential equation by separation of variables.
$\frac{d y}{d x}=\frac{x y+2 y-x-2}{x y-3 y+x-3}$

Darshan Maheshwari
Darshan Maheshwari
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01:31

Problem 21

Solve the given initial-value problem.
$\frac{d y}{d x}=\frac{1}{(x y)^{2}}, \quad y(1)=3$

Darshan Maheshwari
Darshan Maheshwari
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01:31

Problem 22

Solve the given initial-value problem.
$\frac{d y}{d x}=\frac{2 x+\sec ^{2} x}{2 y}, \quad y(0)=-2$

Darshan Maheshwari
Darshan Maheshwari
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01:54

Problem 23

Solve the given initial-value problem.
$\frac{d x}{d t}=4\left(x^{2}+1\right), \quad x(\pi / 4)=1$

Darshan Maheshwari
Darshan Maheshwari
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04:01

Problem 24

Solve the given initial-value problem.
$\frac{d y}{d x}=\frac{y^{2}-1}{x^{2}-1}, \quad y(2)=2$

Darshan Maheshwari
Darshan Maheshwari
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02:52

Problem 25

Solve the given initial-value problem.
$x^{2} \frac{d y}{d x}=y-x y, \quad y(-1)=-1$

Victoria Dollar
Victoria Dollar
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02:20

Problem 26

Solve the given initial-value problem.
$\frac{d y}{d t}+2 y=1, \quad y(0)=\frac{5}{2}$

Darshan Maheshwari
Darshan Maheshwari
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02:47

Problem 27

Solve the given initial-value problem. Write the solution as an explicit algebraic function $y=f(x)$ (see Notes From the Classroom in Section 1.3). You may have to use a trigonometric identity.
$\sqrt{1-x^{2}} \frac{d y}{d x}=\sqrt{1-y^{2}}, \quad y(0)=\frac{\sqrt{3}}{2}$

Darshan Maheshwari
Darshan Maheshwari
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04:31

Problem 28

Solve the given initial-value problem. Write the solution as an explicit algebraic function $y=f(x)$ (see Notes From the Classroom in Section 1.3). You may have to use a trigonometric identity.
$\left(1+x^{4}\right) \frac{d y}{d x}+x+4 x y^{2}=0, \quad y(1)=0$

Darshan Maheshwari
Darshan Maheshwari
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02:36

Problem 29

Use the concept that $y=k$ on $(-\infty, \infty)$ is a constant function if and only if $d y / d x=0$ to determine whether the given differential equation possesses constant solutions. Solve the given differential equation. Assume that $k$ is a real number.
$x \frac{d y}{d x}+6 y=18$

Darshan Maheshwari
Darshan Maheshwari
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01:19

Problem 30

Use the concept that $y=k$ on $(-\infty, \infty)$ is a constant function if and only if $d y / d x=0$ to determine whether the given differential equation possesses constant solutions. Solve the given differential equation. Assume that $k$ is a real number.
$2 \frac{d y}{d x}=5 y+40$

Darshan Maheshwari
Darshan Maheshwari
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01:19

Problem 31

Use the concept that $y=k$ on $(-\infty, \infty)$ is a constant function if and only if $d y / d x=0$ to determine whether the given differential equation possesses constant solutions. Solve the given differential equation. Assume that $k$ is a real number.
$2 \frac{d y}{d x}=5 y+40$

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
01:37

Problem 32

Use the concept that $y=k$ on $(-\infty, \infty)$ is a constant function if and only if $d y / d x=0$ to determine whether the given differential equation possesses constant solutions. Solve the given differential equation. Assume that $k$ is a real number.
$x \frac{d y}{d x}=y^{2}+2 y+4$

Darshan Maheshwari
Darshan Maheshwari
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00:48

Problem 33

Proceed as in Problems $29-32$ to determine whether the given differential equation possesses constant solutions. Solve the given differential equation and then find a solution whose graph passes through the indicated point.
$x \frac{d y}{d x}=y^{2}-y$
(a) (0,1)
(b) (0,0)
(c) $\left(\frac{1}{2}, \frac{1}{2}\right)$

Audrey Fong
Audrey Fong
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00:39

Problem 34

Proceed as in Problems $29-32$ to determine whether the given differential equation possesses constant solutions. Solve the given differential equation and then find a solution whose graph passes through the indicated point.
$\frac{d y}{d x}=y^{2}-9$
(a) (0,0)
(b) (0,3)
(c) $\left(\frac{1}{3}, 1\right)$

Audrey Fong
Audrey Fong
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01:24

Problem 35

Without solving, explain why the initial-value problem
$$\frac{d y}{d x}=\sqrt{y}, \quad y\left(x_{0}\right)=y_{0}$$
has no solution for $y_{0}<0$.

Darshan Maheshwari
Darshan Maheshwari
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Problem 36

A solution of a differential equation that is not a member of the family of solutions of the equation is called a singular solution. Reexamine Problems $29,31,33,$ and 34 and find any singular solutions. In Example $6,$ what would be the solution of the IVP if the initial condition were changed to $y(0)=1 ?$

Victor Salazar
Victor Salazar
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01:38

Problem 37

In Example 3 , it was stated that the solution $y=-\sqrt{25-x^{2}}$ is defined on the open interval $(-5,5) .$ Why would it be incorrect to say that the solution is defined on the closed interval [-5,5]$?$

Willis James
Willis James
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