Question 1

Solve the differential equation. $$x ^ { 2 } \frac { d y } { d x } - y = 4 x ^ { 3 } e ^ { - 1 / x }$$

Accepted Answer

The answer of Solve the differential equation. \[x ^ {...

Question 2

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 25 cells. Find the number of cells after 5 hours.

Accepted Answer

The answer of A common inhabitant of human intestines is...

Question 3

A sum of $\$ 5,000$ is invested at $30 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$.

Accepted Answer

The answer of A sum of $\$ 5,000$ is invested...

Question 4

Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Suppose that a roast turkey is taken from an oven when its temperature has reached $160 ^ { \circ } \mathrm { F }$ and is placed on a table in a room where the temperature is $60 ^ { \circ } \mathrm { F }$. If $u ( t )$ is the temperature of the turkey after $t$ minutes, then Newton's Law of Cooling implies that
$$\frac { d u } { d t } = k ( u - 60 )$$
This could be solved as a separable differential equation. Another method is to make the change of variable $y = u - 60$. If the temperature of the turkey is $150 ^ { \circ } \mathrm { F }$ after half an hour, what is the temperature after $35 \mathrm {~min}$ ?

Accepted Answer

A)  $t = 143 ^ { \circ } \mathrm { F }$ 
B)  $t = 148 ^ { \circ } \mathrm { F }$ 
C)  $t = 298 ^ { \circ } \mathrm { F }$ 
D)  none of these 
E)  $t = 95 ^ { \circ } \mathrm { F }$ 
A)  $t = 143 ^ { \circ } \mathrm { F }$ 
B)  $t = 148 ^ { \circ } \mathrm { F }$ 
C)  $t = 298 ^ { \circ } \mathrm { F }$ 
D)  none of these 
E)  $t = 95 ^ { \circ } \mathrm { F }$

Question 5

Solve the initial-value problem.
$$\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 3$$

Accepted Answer

The answer of Solve the initial-value problem.
\[\frac { d r...

Question 6

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows:
$$\frac { d A } { d t } = 3 A ( 1 - 0.0002 A ) - 0.01 A L$$
$$\frac { d L } { d t } = - 0.4 L + 0.0002 A L$$
Find the equilibrium solution.

Accepted Answer

The answer of We modeled populations of aphids and ladybugs...

Question 7

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 75 cells. Find the number of cells after 2 hours.

Accepted Answer

A)  $P = 16,800$ 
B)  $P = 104,800$ 
C)  $P = 144,800$ 
D)  $P = 4,800$ 
E)  $P = 124,800$ 
A)  $P = 16,800$ 
B)  $P = 104,800$ 
C)  $P = 144,800$ 
D)  $P = 4,800$ 
E)  $P = 124,800$

Question 8

$$\text { Kirchhoff's Law gives us the derivative equation } Q ^ { t } = 12 - 4 Q$$ $$\text { If } Q ( 0 ) = 0 \text {, use Euler's method with step size } 0.1 \text { to estimate } Q \text { after } 0.3 \text { second. }$$

Accepted Answer

The answer of \[\text { Kirchhoff's Law gives us the...

Question 9

Solve the differential equation.
$$\frac { d u } { d t } = 22 + 2 u + 11 t + u t$$

Accepted Answer

The answer of Solve the differential equation.
\[\frac { d u...

Question 10

The population of the world was about $5.3$ billion in 1990 . Birth rates in the 1990 s range from 35 to 40 million per year and death rates range from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. Use the logistic model to predict the world population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the initial population is small compared to the carrying capacity, you can take $k$ to be an estimate of the initial relative growth rate.)

Accepted Answer

A)  $78.3$ billion 
B)  $27.0$ billion 
C)  $17.1$ billion 
D)  $59.2$ billion 
E)  $32.9$ billion 
A)  $78.3$ billion 
B)  $27.0$ billion 
C)  $17.1$ billion 
D)  $59.2$ billion 
E)  $32.9$ billion

Question 11

A phase trajectory is shown for populations of rabbits $( R )$ and foxes $( F )$. Describe how each population changes as time goes by.

Select the correct statement.

Accepted Answer

A)  At $t = B$ the number of rabbits rebounds to 500 . 
B)  At $t = B$ the number of foxes reaches a maximum of about 2400 . 
C)  At $t = B$ the population of foxes reaches a minimum of about 30 . 
A)  At $t = B$ the number of rabbits rebounds to 500 . 
B)  At $t = B$ the number of foxes reaches a maximum of about 2400 . 
C)  At $t = B$ the population of foxes reaches a minimum of about 30 .

Question 12

For what values of $k$ does the function $y = \cos k t$ satisfy the differential equation $49 y ^ { \prime \prime } = - 81 y$ ?

Accepted Answer

A) 
$$k = - \frac { 1 } { 9 }$$ 
B)  $k = \frac { 1 } { 9 }$ 
C) 
$$k = - \frac { 9 } { 7 }$$ 
D)  $k = \frac { 1 } { 7 }$ 
E)  $k = \frac { 9 } { 7 }$ 
A) 
$$k = - \frac { 1 } { 9 }$$ 
B)  $k = \frac { 1 } { 9 }$ 
C) 
$$k = - \frac { 9 } { 7 }$$ 
D)  $k = \frac { 1 } { 7 }$ 
E)  $k = \frac { 9 } { 7 }$

Question 13

Solve the differential equation.
$$x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$$

Accepted Answer

The answer of Solve the differential equation.
\[x ^ { 2...

Question 14

Solve the initial-value problem.
$( 1 + \cos x ) \frac { d y } { d x } = \left( 6 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$

Accepted Answer

A)  $y = \frac { \ln ( 6 + \cos x ) } { \ln ( 6 - \cos x ) }$ 
B) 
$y = \ln \left( \frac { 1 } { 6 } \left( \sec ^ { 12 } \left( \frac { x } { 2 } \right) - 1 \right) \right.$ 
C)  $y = \ln ( 6 - \cos x ) ( 6 + \cos x )$ 
D) 
$y = \ln \left( \frac { 1 } { 6 } \left( 7 \sec ^ { 12 } \left( \frac { x } { 2 } \right) - 1 \right) \right.$ 
E) 
$y = \ln \left( 6 \sec \left( \frac { x } { 2 } \right) - 1 \right)$ 
A)  $y = \frac { \ln ( 6 + \cos x ) } { \ln ( 6 - \cos x ) }$ 
B) 
$y = \ln \left( \frac { 1 } { 6 } \left( \sec ^ { 12 } \left( \frac { x } { 2 } \right) - 1 \right) \right.$ 
C)  $y = \ln ( 6 - \cos x ) ( 6 + \cos x )$ 
D) 
$y = \ln \left( \frac { 1 } { 6 } \left( 7 \sec ^ { 12 } \left( \frac { x } { 2 } \right) - 1 \right) \right.$ 
E) 
$y = \ln \left( 6 \sec \left( \frac { x } { 2 } \right) - 1 \right)$

Question 15

Choose the differential equation corresponding to this direction field. Select the correct answer.

Accepted Answer

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

Question 16

A population is modeled by the differential equation.
$$\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4500 } \right)$$
For what values of $P$ is the population increasing?

Accepted Answer

The answer of A population is modeled by the differential...

Question 17

Solve the initial-value problem.
$$x y ^ { t} = y + x ^ { 2 } \sin x , y ( 11 \pi ) = 0$$

Accepted Answer

The answer of Solve the initial-value problem.
\[x y ^ {...

Question 18

A population is modeled by the differential equation.
$$\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4560 } \right)$$
For what values of $P$ is the population increasing?

Accepted Answer

A) $P > 1.4$ B) $P > 4560$ C) $P < 4580$ D) $0 < P < 4560$ E) $0 < P < 1.4$ A) $P > 1.4$ B) $P > 4560$ C) $P < 4580$ D) $0 < P < 4560$ E) $0 < P < 1.4$

Question 19

Solve the initial-value problem.
$$\frac { d u } { d t } = \frac { 2 t + \sec ^ { 2 } t } { 2 u } , u ( 0 ) = 4$$

Accepted Answer

The answer of Solve the initial-value problem.
\[\frac { d u...

Question 20

The population of the world was about $5.3$ billion in 1990 . Birth rates in the 1990 s range from 35 to 40 million per year and death rates range from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. Use the logistic model to predict the world population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the initial population is small compared to the carrying capacity, you can take $k$ to be an estimate of the initial relative growth rate.) Select the correct answer.

Accepted Answer

A)  $78.3$ billion 
B)  $27.0$ billion 
C)  $17.1$ billion 
D)  $59.2$ billion 
E)  $32.9$ billion 
A)  $78.3$ billion 
B)  $27.0$ billion 
C)  $17.1$ billion 
D)  $59.2$ billion 
E)  $32.9$ billion

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 4 x ^ { 3 } e ^ { - 1 / x }$

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 25 cells. Find the number of cells after 5 hours.

A sum of $\$ 5,000$ is invested at $30 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Solve the initial-value problem. $\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 3$

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: $\frac { d A } { d t } = 3 A ( 1 - 0.0002 A ) - 0.01 A L$ $\frac { d L } { d t } = - 0.4 L + 0.0002 A L$ Find the equilibrium solution.

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 75 cells. Find the number of cells after 2 hours.

$\text { Kirchhoff's Law gives us the derivative equation } Q ^ { t } = 12 - 4 Q$ $\text { If } Q ( 0 ) = 0 \text {, use Euler's method with step size } 0.1 \text { to estimate } Q \text { after } 0.3 \text { second. }$

Solve the differential equation. $\frac { d u } { d t } = 22 + 2 u + 11 t + u t$

A phase trajectory is shown for populations of rabbits $( R )$ and foxes $( F )$ . Describe how each population changes as time goes by. Select the correct statement.

For what values of $k$ does the function $y = \cos k t$ satisfy the differential equation $49 y ^ { \prime \prime } = - 81 y$ ?

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

Solve the initial-value problem. $( 1 + \cos x ) \frac { d y } { d x } = \left( 6 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$

Choose the differential equation corresponding to this direction field. Select the correct answer.

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4500 } \right)$ For what values of $P$ is the population increasing?

Solve the initial-value problem. $x y ^ { t} = y + x ^ { 2 } \sin x , y ( 11 \pi ) = 0$

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4560 } \right)$ For what values of $P$ is the population increasing?

Solve the initial-value problem. $\frac { d u } { d t } = \frac { 2 t + \sec ^ { 2 } t } { 2 u } , u ( 0 ) = 4$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Exam 9: Differential Equations

Solve the differential equation. x2dydx−y=4x3e−1/xx ^ { 2 } \frac { d y } { d x } - y = 4 x ^ { 3 } e ^ { - 1 / x }x2dxdy​−y=4x3e−1/x

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 25 cells. Find the number of cells after 5 hours.

A sum of $5,000\$ 5,000$5,000 is invested at 30%30 \%30% interest. If A(t)A ( t )A(t) is the amount of the investment at time ttt for the case of continuous compounding, write a differential equation and an initial condition satisfied by A(t)A ( t )A(t) .

Solve the initial-value problem. drdt+2tr−r=0,r(0)=3\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 3dtdr​+2tr−r=0,r(0)=3

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 75 cells. Find the number of cells after 2 hours.

Solve the differential equation. dudt=22+2u+11t+ut\frac { d u } { d t } = 22 + 2 u + 11 t + u tdtdu​=22+2u+11t+ut

A phase trajectory is shown for populations of rabbits (R)( R )(R) and foxes (F)( F )(F) . Describe how each population changes as time goes by. Select the correct statement.

For what values of kkk does the function y=cos⁡kty = \cos k ty=coskt satisfy the differential equation 49y′′=−81y49 y ^ { \prime \prime } = - 81 y49y′′=−81y ?

Solve the differential equation. x2dydx−y=2x3e−1/xx ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }x2dxdy​−y=2x3e−1/x

Solve the initial-value problem. (1+cos⁡x)dydx=(6+e−y)sin⁡x,y(0)=0( 1 + \cos x ) \frac { d y } { d x } = \left( 6 + e ^ { - y } \right) \sin x , y ( 0 ) = 0(1+cosx)dxdy​=(6+e−y)sinx,y(0)=0

Choose the differential equation corresponding to this direction field. Select the correct answer.

A population is modeled by the differential equation. dPdt=1.4P(1−P4500)\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4500 } \right)dtdP​=1.4P(1−4500P​) For what values of PPP is the population increasing?

Solve the initial-value problem. xyt=y+x2sin⁡x,y(11π)=0x y ^ { t} = y + x ^ { 2 } \sin x , y ( 11 \pi ) = 0xyt=y+x2sinx,y(11π)=0

A population is modeled by the differential equation. dPdt=1.4P(1−P4560)\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4560 } \right)dtdP​=1.4P(1−4560P​) For what values of PPP is the population increasing?

Solve the initial-value problem. dudt=2t+sec⁡2t2u,u(0)=4\frac { d u } { d t } = \frac { 2 t + \sec ^ { 2 } t } { 2 u } , u ( 0 ) = 4dtdu​=2u2t+sec2t​,u(0)=4

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 4 x ^ { 3 } e ^ { - 1 / x }$

A sum of $\$ 5,000$ is invested at $30 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Solve the initial-value problem. $\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 3$

Solve the differential equation. $\frac { d u } { d t } = 22 + 2 u + 11 t + u t$

A phase trajectory is shown for populations of rabbits $( R )$ and foxes $( F )$ . Describe how each population changes as time goes by. Select the correct statement.

For what values of $k$ does the function $y = \cos k t$ satisfy the differential equation $49 y ^ { \prime \prime } = - 81 y$ ?

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

Solve the initial-value problem. $( 1 + \cos x ) \frac { d y } { d x } = \left( 6 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4500 } \right)$ For what values of $P$ is the population increasing?

Solve the initial-value problem. $x y ^ { t} = y + x ^ { 2 } \sin x , y ( 11 \pi ) = 0$

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4560 } \right)$ For what values of $P$ is the population increasing?

Solve the initial-value problem. $\frac { d u } { d t } = \frac { 2 t + \sec ^ { 2 } t } { 2 u } , u ( 0 ) = 4$