Question 1

Use integration to find a general solution of the differential equation $\frac { d y } { d x } = 13 x \cos \left( 8 x ^ { 2 } \right)$

Accepted Answer

A)  $y = \frac { 13 x \sin \left( 16 x ^ { 2 } \right) } { 2 x } + C$ 
B)  $y = \frac { 13 x \cos \left( 8 x ^ { 2 } \right) } { 16 } + C$ 
C)  $y = \frac { 13 \sin \left( 8 x ^ { 2 } \right) } { 16 } + C$ 
D)  $y = \frac { 13 \cos \left( 8 x ^ { 2 } \right) } { 16 x } + C$ 
E)  $y = \frac { 13 \sin \left( 16 x ^ { 2 } \right) } { 4 } + C$ 
A)  $y = \frac { 13 x \sin \left( 16 x ^ { 2 } \right) } { 2 x } + C$ 
B)  $y = \frac { 13 x \cos \left( 8 x ^ { 2 } \right) } { 16 } + C$ 
C)  $y = \frac { 13 \sin \left( 8 x ^ { 2 } \right) } { 16 } + C$ 
D)  $y = \frac { 13 \cos \left( 8 x ^ { 2 } \right) } { 16 x } + C$ 
E)  $y = \frac { 13 \sin \left( 16 x ^ { 2 } \right) } { 4 } + C$

Question 2

$\text { Find the particular solution of the differential equation } \frac { d r } { d s } = e ^ { y - 7 s } \text { that satisfies the }$ initial condition $r ( 0 ) = 0$.

Accepted Answer

A)  $r = \ln \left( 7 + e ^ { - 7 s } \right) + C$ 
B)  $r = \ln ( 7 ) - \ln \left( 6 + e ^ { - 7 s } \right)$ 
C)  $r = e ^ { \gamma - 7 s }$ 
D) $r = \ln \left( \frac { 8 + e ^ { - 7 s } } { 7 } \right)$ 
E) $r = \left( 1 + e ^ { - 7 s } \right) ^ { 7 }$ 
A)  $r = \ln \left( 7 + e ^ { - 7 s } \right) + C$ 
B)  $r = \ln ( 7 ) - \ln \left( 6 + e ^ { - 7 s } \right)$ 
C)  $r = e ^ { \gamma - 7 s }$ 
D) $r = \ln \left( \frac { 8 + e ^ { - 7 s } } { 7 } \right)$ 
E) $r = \left( 1 + e ^ { - 7 s } \right) ^ { 7 }$

Question 3

The initial investment in a savings account in which interest is compounded continuously is $\$ 604$. If the time required to double the amount is $9 \frac { 1 } { 2 }$ years, what is the amount after 15 years? Round your answer to the nearest cent.

Accepted Answer

A)  $\$ 1,917.58$ 
B)  $\$ 1,804.46$ 
C)  $\$ 1,907.37$ 
D)  $\$ 1,404.46$ 
E)  $\$ 8,278.18$ 
A)  $\$ 1,917.58$ 
B)  $\$ 1,804.46$ 
C)  $\$ 1,907.37$ 
D)  $\$ 1,404.46$ 
E)  $\$ 8,278.18$

Question 4

$\text { A calf that weighs } 80 \text { pounds at birth gains weight at the rate } \frac { d w } { d t } = k ( 1500 - w ) \text {, }$ where $w$ is weight in pounds and $t$ is time in years. What is the maximum weight of the animal if one uses the model $w = 1500 - 1420 e ^ { - 0.9 t }$ ?

Accepted Answer

A)  $1500 \mathrm { lb }$ 
B)  $880 \mathrm { lb }$ 
C)  $1420 \mathrm { lb }$ 
D)  $1580 \mathrm { lb }$ 
E)  $2300 \mathrm { lb }$ 
A)  $1500 \mathrm { lb }$ 
B)  $880 \mathrm { lb }$ 
C)  $1420 \mathrm { lb }$ 
D)  $1580 \mathrm { lb }$ 
E)  $2300 \mathrm { lb }$

Question 5

$\text { Which of the following is a solution of the differential equation } x y ^ { t } - 4 y = x ^ { 5 } e ^ { x } \text { ? }$

Accepted Answer

A) $y = 4 x ^ { 5 } e ^ { 2 x }$ 
B)  $y = 6 e ^ { 2 x } - 7 \sin 2 x$ 
C)  $y = x ^ { 4 } e ^ { x }$ 
D)  $y = 5 e ^ { - 2 x }$ 
E)  $y = \ln x$ 
A) $y = 4 x ^ { 5 } e ^ { 2 x }$ 
B)  $y = 6 e ^ { 2 x } - 7 \sin 2 x$ 
C)  $y = x ^ { 4 } e ^ { x }$ 
D)  $y = 5 e ^ { - 2 x }$ 
E)  $y = \ln x$

Question 6

$\text { Find the general solution of the differential equation } \frac { d y } { d x } = \frac { 5 x ^ { 2 } } { 8 y ^ { 2 } } \text {. }$

Accepted Answer

A)  $y = \frac { 5 } { 8 } x ^ { 3 } + C ^ { \prime }$ 
B)  $y = \sqrt [ 3 ] { \frac { 5 } { 8 } x ^ { 3 } + C }$ 
C)  $y = \sqrt { \frac { 8 } { 5 } x ^ { 3 } + C }$ 
D)  $y = \sqrt [ 3 ] { \frac { x ^ { 2 } } { 8 } + C }$ 
E)  $y = \sqrt [ 3 ] { 5 x ^ { 3 } + 8 C }$ 
A)  $y = \frac { 5 } { 8 } x ^ { 3 } + C ^ { \prime }$ 
B)  $y = \sqrt [ 3 ] { \frac { 5 } { 8 } x ^ { 3 } + C }$ 
C)  $y = \sqrt { \frac { 8 } { 5 } x ^ { 3 } + C }$ 
D)  $y = \sqrt [ 3 ] { \frac { x ^ { 2 } } { 8 } + C }$ 
E)  $y = \sqrt [ 3 ] { 5 x ^ { 3 } + 8 C }$

Question 7

The half life of the radium isotope Ra-226 is approximately 1,599 years. If the amount left after 1,000 years is 1.8 g, what is the amount after 2000 years? Round your answer to Three decimal places.

Accepted Answer

A) 1.167 g 
B) 0.939 g 
C) 1.800 g 
D) 0.490 g 
E) 2.334 g 
A) 1.167 g 
B) 0.939 g 
C) 1.800 g 
D) 0.490 g 
E) 2.334 g

Question 8

$\text { Solve the differential equation } y ^ { t } = \frac { \sqrt { x } } { 2 y } \text {. }$

Accepted Answer

A) $2 y ^ { 2 } = 4 x ^ { \frac { 3 } { 2 } } + C$ 
B) $6 y ^ { 2 } = 4 x ^ { \frac { 3 } { 2 } } + C$ 
C)  $6 y ^ { 2 } = 2 x ^ { \frac { 3 } { 2 } } + C$ 
D) $4 y ^ { 2 } = 6 x ^ { \frac { 3 } { 2 } } + C$ 
E) $4 y ^ { 2 } = 2 x ^ { \frac { 3 } { 2 } } + C$ 
A) $2 y ^ { 2 } = 4 x ^ { \frac { 3 } { 2 } } + C$ 
B) $6 y ^ { 2 } = 4 x ^ { \frac { 3 } { 2 } } + C$ 
C)  $6 y ^ { 2 } = 2 x ^ { \frac { 3 } { 2 } } + C$ 
D) $4 y ^ { 2 } = 6 x ^ { \frac { 3 } { 2 } } + C$ 
E) $4 y ^ { 2 } = 2 x ^ { \frac { 3 } { 2 } } + C$

Question 9

Find the particular solution of the differential equation $x d y = ( x + y + 6 ) d x \text { that }$ satisfies the boundary condition $y ( 1 ) = 5$.

Accepted Answer

A)  $y = 6 x \ln | x | - 10 x - 5$ 
B)  $y = x \ln | x | + 11 x - 6$ 
C)  $y = x \ln | x | + 11 x + 5$ 
D)  $y = 6 x \ln | x | + 10 x - 5$ 
E)  $y = x \ln | x | - 11 x + 6$ 
A)  $y = 6 x \ln | x | - 10 x - 5$ 
B)  $y = x \ln | x | + 11 x - 6$ 
C)  $y = x \ln | x | + 11 x + 5$ 
D)  $y = 6 x \ln | x | + 10 x - 5$ 
E)  $y = x \ln | x | - 11 x + 6$

Question 10

$\text { Solve the homogeneous differential equation } y ^ { t } = \frac { 5 x + 9 y } { x }$

Accepted Answer

A)  $y = \ln ( 5 x + 9 ) + C$ 
B)  $y = - \frac { 9 } { 4 } x - x ^ { 18 } + C$ 
C)  $y = \frac { 5 } { 6 } x ^ { 9 } + C$ 
D)  $y = \frac { 5 } { 9 } x + C \cdot x ^ { 5 }$ 
E)  $y = - \frac { 5 } { 8 } x + C \cdot x ^ { 9 }$ 
A)  $y = \ln ( 5 x + 9 ) + C$ 
B)  $y = - \frac { 9 } { 4 } x - x ^ { 18 } + C$ 
C)  $y = \frac { 5 } { 6 } x ^ { 9 } + C$ 
D)  $y = \frac { 5 } { 9 } x + C \cdot x ^ { 5 }$ 
E)  $y = - \frac { 5 } { 8 } x + C \cdot x ^ { 9 }$

Question 11

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. $\frac { d y } { d x } = 1 - y$

Accepted Answer

A)  $y = C \ln ( 1 - y )$ 
B)  $y = 1 x + C e ^ { - x }$ 
C)  $y = 1 - 2 C e ^ { x }$ 
D)  $y = C \ln ( y - 1 )$ 
E)  $y = 1 + C e ^ { - x }$ 
A)  $y = C \ln ( 1 - y )$ 
B)  $y = 1 x + C e ^ { - x }$ 
C)  $y = 1 - 2 C e ^ { x }$ 
D)  $y = C \ln ( y - 1 )$ 
E)  $y = 1 + C e ^ { - x }$

Question 12

A conservation organization releases 30 panthers into a preserve. After 3 years, there are 50 panthers in the preserve. The preserve has a carrying capacity of 150. Determine the time it Takes for the population to reach 110.

Accepted Answer

A) 13.139 years 
B) 8.994 years 
C) 10.378 years 
D) 7.811 years 
E) 12.003 years 
A) 13.139 years 
B) 8.994 years 
C) 10.378 years 
D) 7.811 years 
E) 12.003 years

Question 13

$\text { The logistic function } P ( t ) = \frac { 20 } { 1 + 0.5 e ^ { - 0.2 t } } \text { models the growth of a population. }$ Identify the value of k.

Accepted Answer

A) 1.7 
B) 2.2 
C) 0.2 
D) 20 
E) 0.5 
A) 1.7 
B) 2.2 
C) 0.2 
D) 20 
E) 0.5

Question 14

Find the orthogonal trajectories of the family $y = C e ^ { 8 x }$

Accepted Answer

A)  $8 y ^ { 2 } = - 2 x + C$ 
B)  $\ln y = 8 x + C$ 
C)  $y = 8 C e ^ { 8 x }$ 
D)  $y = k e ^ { - 8 x }$ 
E)  $y = C \ln ( 8 x )$ 
A)  $8 y ^ { 2 } = - 2 x + C$ 
B)  $\ln y = 8 x + C$ 
C)  $y = 8 C e ^ { 8 x }$ 
D)  $y = k e ^ { - 8 x }$ 
E)  $y = C \ln ( 8 x )$

Question 15

Select from the choices below the slope field for the differential equation. $\frac { d y } { d x } = 2 \sin x$

Accepted Answer

A) 
  
B) 
  
C) 
  
D)  none of the above 
A) 
  
B) 
  
C) 
  
D)  none of the above

Question 16

The half-life of the radium isotope Ra-226 is approximately 1,599 years. What percent of a given amount remains after 800 years? Round your answer to two decimal places.

Accepted Answer

A) 70.70 % 
B) 5.71 % 
C) 72.70 % 
D) 25.02 % 
E) 0.71 % 
A) 70.70 % 
B) 5.71 % 
C) 72.70 % 
D) 25.02 % 
E) 0.71 %

Question 17

$\text { Use integration to find a general solution of the differential equation } \frac { d y } { d x } = \frac { 3 x } { 3 + x ^ { 2 } } \text {. }$

Accepted Answer

A)  $y = \frac { 3 } { 2 } \ln \left( \left| 3 + x ^ { 2 } \right| \right) + C$ 
B)  $y = \frac { 3 } { 2 x } \ln \left( \left| 6 + x ^ { 2 } \right| \right) + C$ 
C)  $y = \frac { 6 } { x ^ { 2 } } \ln \left( \left| 3 + x ^ { 2 } \right| \right) + C$ 
D)  $y = \frac { 3 } { x \ln \left( \left| 3 + x ^ { 2 } \right| \right) } + C$ 
E) $y = \frac { 3 x ^ { 2 } } { \ln \left( \left| 3 + x ^ { 2 } \right| \right) } + C$ 
A)  $y = \frac { 3 } { 2 } \ln \left( \left| 3 + x ^ { 2 } \right| \right) + C$ 
B)  $y = \frac { 3 } { 2 x } \ln \left( \left| 6 + x ^ { 2 } \right| \right) + C$ 
C)  $y = \frac { 6 } { x ^ { 2 } } \ln \left( \left| 3 + x ^ { 2 } \right| \right) + C$ 
D)  $y = \frac { 3 } { x \ln \left( \left| 3 + x ^ { 2 } \right| \right) } + C$ 
E) $y = \frac { 3 x ^ { 2 } } { \ln \left( \left| 3 + x ^ { 2 } \right| \right) } + C$

Question 18

The isotope ${ } ^ { 14 } \mathrm { C }$ has a half-life of 5,715 years. Given an initial amount of 11 grams of the isotope, how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places.

Accepted Answer

A)  $7.2469 \mathrm { gm } , 4.1988 \mathrm { gm }$ 
B)  $6.2117 \mathrm { gm } , 3.5989 \mathrm { gm }$ 
C)  $10.3528 \mathrm { gm } , 5.9982 \mathrm { gm }$ 
D)  $4.1411 \mathrm { gm } , 2.3993 \mathrm { gm }$ 
E)  $12.4233 \mathrm { gm } , 7.1979 \mathrm { gm }$ 
A)  $7.2469 \mathrm { gm } , 4.1988 \mathrm { gm }$ 
B)  $6.2117 \mathrm { gm } , 3.5989 \mathrm { gm }$ 
C)  $10.3528 \mathrm { gm } , 5.9982 \mathrm { gm }$ 
D)  $4.1411 \mathrm { gm } , 2.3993 \mathrm { gm }$ 
E)  $12.4233 \mathrm { gm } , 7.1979 \mathrm { gm }$

Question 19

A 300-gallon tank is full of a solution containing 35 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred Solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function Of t.

Accepted Answer

A) $Q = e ^ { ( 1 / 30 ) t }$ 
B)  $Q = 30 e ^ { - ( 1 / 10 ) t }$ 
C)  $Q = 300 e ^ { - ( 1 / 10 ) z }$ 
D)  $Q = 35 e ^ { - ( 1 / 10 ) t }$ 
E)  $Q = e ^ { - ( 1 / 35 ) z }$ 
A) $Q = e ^ { ( 1 / 30 ) t }$ 
B)  $Q = 30 e ^ { - ( 1 / 10 ) t }$ 
C)  $Q = 300 e ^ { - ( 1 / 10 ) z }$ 
D)  $Q = 35 e ^ { - ( 1 / 10 ) t }$ 
E)  $Q = e ^ { - ( 1 / 35 ) z }$

Question 20

$\text { Find the particular solution of the differential equation } 3 x + 20 y y ^ { t } = 0 \text { that satisfies }$ the initial condition $y = 5$ when $x = 2$, where $3 x ^ { 2 } + 20 y ^ { 2 } = C$ is the general solution.

Accepted Answer

A)  $3 x ^ { 2 } + 20 y ^ { 2 } = 504$ 
B)  $3 x ^ { 2 } + 20 y ^ { 2 } = 155$ 
C)  $3 x ^ { 2 } + 20 y ^ { 2 } = 112$ 
D)  $3 x ^ { 2 } + 20 y ^ { 2 } = 37$ 
E)  $3 x ^ { 2 } + 20 y ^ { 2 } = 512$ 
A)  $3 x ^ { 2 } + 20 y ^ { 2 } = 504$ 
B)  $3 x ^ { 2 } + 20 y ^ { 2 } = 155$ 
C)  $3 x ^ { 2 } + 20 y ^ { 2 } = 112$ 
D)  $3 x ^ { 2 } + 20 y ^ { 2 } = 37$ 
E)  $3 x ^ { 2 } + 20 y ^ { 2 } = 512$

Use integration to find a general solution of the differential equation $\frac { d y } { d x } = 13 x \cos \left( 8 x ^ { 2 } \right)$

$\text { Find the particular solution of the differential equation } \frac { d r } { d s } = e ^ { y - 7 s } \text { that satisfies the }$ initial condition $r ( 0 ) = 0$ .

The initial investment in a savings account in which interest is compounded continuously is $\$ 604$ . If the time required to double the amount is $9 \frac { 1 } { 2 }$ years, what is the amount after 15 years? Round your answer to the nearest cent.

$\text { A calf that weighs } 80 \text { pounds at birth gains weight at the rate } \frac { d w } { d t } = k ( 1500 - w ) \text {, }$ where $w$ is weight in pounds and $t$ is time in years. What is the maximum weight of the animal if one uses the model $w = 1500 - 1420 e ^ { - 0.9 t }$ ?

$\text { Which of the following is a solution of the differential equation } x y ^ { t } - 4 y = x ^ { 5 } e ^ { x } \text { ? }$

$\text { Find the general solution of the differential equation } \frac { d y } { d x } = \frac { 5 x ^ { 2 } } { 8 y ^ { 2 } } \text {. }$

The half life of the radium isotope Ra-226 is approximately 1,599 years. If the amount left after 1,000 years is 1.8 g, what is the amount after 2000 years? Round your answer to Three decimal places.

$\text { Solve the differential equation } y ^ { t } = \frac { \sqrt { x } } { 2 y } \text {. }$

Find the particular solution of the differential equation $x d y = ( x + y + 6 ) d x \text { that }$ satisfies the boundary condition $y ( 1 ) = 5$ .

$\text { Solve the homogeneous differential equation } y ^ { t } = \frac { 5 x + 9 y } { x }$

A conservation organization releases 30 panthers into a preserve. After 3 years, there are 50 panthers in the preserve. The preserve has a carrying capacity of 150. Determine the time it Takes for the population to reach 110.

$\text { The logistic function } P ( t ) = \frac { 20 } { 1 + 0.5 e ^ { - 0.2 t } } \text { models the growth of a population. }$ Identify the value of k.

Find the orthogonal trajectories of the family $y = C e ^ { 8 x }$

Select from the choices below the slope field for the differential equation. $\frac { d y } { d x } = 2 \sin x$

The half-life of the radium isotope Ra-226 is approximately 1,599 years. What percent of a given amount remains after 800 years? Round your answer to two decimal places.

$\text { Use integration to find a general solution of the differential equation } \frac { d y } { d x } = \frac { 3 x } { 3 + x ^ { 2 } } \text {. }$

The isotope ${ } ^ { 14 } \mathrm { C }$ has a half-life of 5,715 years. Given an initial amount of 11 grams of the isotope, how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places.

A 300-gallon tank is full of a solution containing 35 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred Solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function Of t.

$\text { Find the particular solution of the differential equation } 3 x + 20 y y ^ { t } = 0 \text { that satisfies }$ the initial condition $y = 5$ when $x = 2$ , where $3 x ^ { 2 } + 20 y ^ { 2 } = C$ is the general solution.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 6: Slope Fields and Eulers Method

Use integration to find a general solution of the differential equation dydx=13xcos⁡(8x2)\frac { d y } { d x } = 13 x \cos \left( 8 x ^ { 2 } \right)dxdy​=13xcos(8x2)

The initial investment in a savings account in which interest is compounded continuously is $604\$ 604$604 . If the time required to double the amount is 9129 \frac { 1 } { 2 }921​ years, what is the amount after 15 years? Round your answer to the nearest cent.

Which of the following is a solution of the differential equation xyt−4y=x5ex ? \text { Which of the following is a solution of the differential equation } x y ^ { t } - 4 y = x ^ { 5 } e ^ { x } \text { ? } Which of the following is a solution of the differential equation xyt−4y=x5ex ?

Find the general solution of the differential equation dydx=5x28y2. \text { Find the general solution of the differential equation } \frac { d y } { d x } = \frac { 5 x ^ { 2 } } { 8 y ^ { 2 } } \text {. } Find the general solution of the differential equation dxdy​=8y25x2​.

The half life of the radium isotope Ra-226 is approximately 1,599 years. If the amount left after 1,000 years is 1.8 g, what is the amount after 2000 years? Round your answer to Three decimal places.

Solve the differential equation yt=x2y. \text { Solve the differential equation } y ^ { t } = \frac { \sqrt { x } } { 2 y } \text {. } Solve the differential equation yt=2yx​​.

Find the particular solution of the differential equation xdy=(x+y+6)dx that x d y = ( x + y + 6 ) d x \text { that }xdy=(x+y+6)dx that satisfies the boundary condition y(1)=5y ( 1 ) = 5y(1)=5 .

Solve the homogeneous differential equation yt=5x+9yx\text { Solve the homogeneous differential equation } y ^ { t } = \frac { 5 x + 9 y } { x } Solve the homogeneous differential equation yt=x5x+9y​

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. dydx=1−y\frac { d y } { d x } = 1 - ydxdy​=1−y

A conservation organization releases 30 panthers into a preserve. After 3 years, there are 50 panthers in the preserve. The preserve has a carrying capacity of 150. Determine the time it Takes for the population to reach 110.

Find the orthogonal trajectories of the family y=Ce8xy = C e ^ { 8 x }y=Ce8x

Select from the choices below the slope field for the differential equation. dydx=2sin⁡x\frac { d y } { d x } = 2 \sin xdxdy​=2sinx

The half-life of the radium isotope Ra-226 is approximately 1,599 years. What percent of a given amount remains after 800 years? Round your answer to two decimal places.

The isotope 14C{ } ^ { 14 } \mathrm { C }14C has a half-life of 5,715 years. Given an initial amount of 11 grams of the isotope, how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places.

A 300-gallon tank is full of a solution containing 35 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred Solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function Of t.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Use integration to find a general solution of the differential equation $\frac { d y } { d x } = 13 x \cos \left( 8 x ^ { 2 } \right)$

The initial investment in a savings account in which interest is compounded continuously is $\$ 604$ . If the time required to double the amount is $9 \frac { 1 } { 2 }$ years, what is the amount after 15 years? Round your answer to the nearest cent.

$\text { Which of the following is a solution of the differential equation } x y ^ { t } - 4 y = x ^ { 5 } e ^ { x } \text { ? }$

$\text { Find the general solution of the differential equation } \frac { d y } { d x } = \frac { 5 x ^ { 2 } } { 8 y ^ { 2 } } \text {. }$

$\text { Solve the differential equation } y ^ { t } = \frac { \sqrt { x } } { 2 y } \text {. }$

Find the particular solution of the differential equation $x d y = ( x + y + 6 ) d x \text { that }$ satisfies the boundary condition $y ( 1 ) = 5$ .

$\text { Solve the homogeneous differential equation } y ^ { t } = \frac { 5 x + 9 y } { x }$

Find the orthogonal trajectories of the family $y = C e ^ { 8 x }$

Select from the choices below the slope field for the differential equation. $\frac { d y } { d x } = 2 \sin x$

The isotope ${ } ^ { 14 } \mathrm { C }$ has a half-life of 5,715 years. Given an initial amount of 11 grams of the isotope, how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places.