Question 1

Solve the differential equation.
$$\frac { d u } { d t } = 15 + 5 u + 3 t + u t$$

Accepted Answer

A) 
$$u = - 3 + C e ^ { \frac { t ^ { 2 } } { 2 } - 5 t }$$ 
B)  $u = - 3 + C e ^ { t ^ { 2 } - 5 t }$ 
C) 
$$u = - 3 + C e ^ { \frac { t ^ { 2 } } { 2 } + 5 t }$$ 
D) 
$$u = - 3 + C e ^ { \frac { t ^ { 2 } } { 2 } + 5 t ^ { 2 } }$$ 
E)  $u = - 3 + C e ^ { t ^ { 2 } + 5 t }$ 
A) 
$$u = - 3 + C e ^ { \frac { t ^ { 2 } } { 2 } - 5 t }$$ 
B)  $u = - 3 + C e ^ { t ^ { 2 } - 5 t }$ 
C) 
$$u = - 3 + C e ^ { \frac { t ^ { 2 } } { 2 } + 5 t }$$ 
D) 
$$u = - 3 + C e ^ { \frac { t ^ { 2 } } { 2 } + 5 t ^ { 2 } }$$ 
E)  $u = - 3 + C e ^ { t ^ { 2 } + 5 t }$

Question 2

A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point $P$ is 3 times the y-coordinate $P$. What is the equation of the curve?

Accepted Answer

A)  $y = 2 e ^ { 3 x + 24 }$ 
B)  $y = \frac { e ^ { 3 x - 24 } } { 2 }$ 
C)  $y = 2 e ^ { 3 x - 8 }$ 
D)  $y = 2 e ^ { 3 x - 24 }$ 
E)  $y = 2 e ^ { x - 24 }$ 
A)  $y = 2 e ^ { 3 x + 24 }$ 
B)  $y = \frac { e ^ { 3 x - 24 } } { 2 }$ 
C)  $y = 2 e ^ { 3 x - 8 }$ 
D)  $y = 2 e ^ { 3 x - 24 }$ 
E)  $y = 2 e ^ { x - 24 }$

Question 3

Solve the initial-value problem.
$$\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 10$$

Accepted Answer

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

Question 4

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows:
$$\frac { d A } { d t } = 2 A ( 1 - 0.0005 A ) - 0.01 A L$$
$$\frac { d L } { d t } = - 0.6 L + 0.0005 A L$$
Find the equilibrium solution.

Accepted Answer

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

Question 5

Suppose that a population grows according to a logistic model with carrying capacity 3,000 and $k = 0.02$ per year. Choose the logistic differential equation for these data.

Accepted Answer

The answer of Suppose that a population grows according to...

Question 6

Solve the differential equation.
$$( 8 + t ) \frac { d u } { d t } + u = 8 + t , t > 0$$

Accepted Answer

The answer of Solve the differential equation.
\[( 8 + t...

Question 7

Suppose that a population grows according to a logistic model with carrying capacity 7,300 and $k = 0.04$ per year. Write the logistic differential equation for these data.

Accepted Answer

The answer of Suppose that a population grows according to...

Question 8

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows:
$$\frac { d A } { d t } = 2 A ( 1 - 0.0005 A ) - 0.01 A L$$
$\frac { d L } { d t } = - 0.6 L + 0.0005 A L$
Find the equilibrium solution.

Accepted Answer

A)  $A = 3200 , L = 110$ 
B)  $A = 4700 , L = 90$ 
C)  $A = 3700 , L = 120$ 
D)  $A = 1200 , L = 80$ 
E)  $A = 1500 , L = 100$ 
A)  $A = 3200 , L = 110$ 
B)  $A = 4700 , L = 90$ 
C)  $A = 3700 , L = 120$ 
D)  $A = 1200 , L = 80$ 
E)  $A = 1500 , L = 100$

Question 9

Let $P ( t )$ be the performance level of someone learning a skill as a function of the training time $t$. The graph of $P$ is called a learning curve. We propose the differential equation
$$\frac { d P } { d t } = r ( G - P ( t ) )$$
as a reasonable model for learning, where $r$ is a positive constant. Solve it as a linear differential equation.

Accepted Answer

A)  $P ( t ) = G - C e ^ { - n }$ 
B)  $P ( t ) = G + C e ^ { - n t }$ 
C)  $P ( t ) = G + C e ^ { n t }$ 
D)  $P ( t ) = G C e ^ { n }$ 
E)  $P ( t ) = G - C e ^ { n }$ 
A)  $P ( t ) = G - C e ^ { - n }$ 
B)  $P ( t ) = G + C e ^ { - n t }$ 
C)  $P ( t ) = G + C e ^ { n t }$ 
D)  $P ( t ) = G C e ^ { n }$ 
E)  $P ( t ) = G - C e ^ { n }$

Question 10

Solve the initial-value problem.
$$x y ^ { t } - y = x \ln x , \quad y ( 1 ) = 4$$

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

Which equation does the function $y = e ^ { - 6 t }$ satisfy? Select the correct answer.

Accepted Answer

A)  $y ^ {t t } + y ^ { t } + 42 y = 0$ 
B)  $y ^ { tt } - 3 y ^ {t } + 42 y = 0$ 
C)  $y ^ { tt } - y ^ { t } - 42 y = 0$ 
D)  $y ^ { tt } + y ^ { t } - 42 y = 0$ 
E)  $y ^ { tt } - y ^ { t } + 42 y = 0$ 
A)  $y ^ {t t } + y ^ { t } + 42 y = 0$ 
B)  $y ^ { tt } - 3 y ^ {t } + 42 y = 0$ 
C)  $y ^ { tt } - y ^ { t } - 42 y = 0$ 
D)  $y ^ { tt } + y ^ { t } - 42 y = 0$ 
E)  $y ^ { tt } - y ^ { t } + 42 y = 0$

Question 13

Determine whether the differential equation is linear.
$$y ^ { t } + 3 x ^ { 2 } y = 6 x ^ { 2 }$$

Accepted Answer

A)  the equation is not linear 
B)  the equation is linear 
A)  the equation is not linear 
B)  the equation is linear

Question 14

Solve the differential equation. Select the correct answer.
$$( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0$$

Accepted Answer

A) 
$u = \frac { t ^ { 2 } + C } { t + 6 }$ 
B) 
$u = 6 t + \frac { t ^ { 2 } } { 6 } + C$ 
C) 
$$u = \frac { 6 t + t ^ { 2 } + C } { t + 6 }$$ 
D) 
$$u = \frac { 6 t + \frac { t ^ { 2 } } { 2 } + C } { t + 6 }$$ 
E) 
$$u = \frac { 6 t + \frac { 1 } { 2 } t ^ { 2 } } { \frac { 3 } { 6 } t + 2 } + C$$ 
A) 
$u = \frac { t ^ { 2 } + C } { t + 6 }$ 
B) 
$u = 6 t + \frac { t ^ { 2 } } { 6 } + C$ 
C) 
$$u = \frac { 6 t + t ^ { 2 } + C } { t + 6 }$$ 
D) 
$$u = \frac { 6 t + \frac { t ^ { 2 } } { 2 } + C } { t + 6 }$$ 
E) 
$$u = \frac { 6 t + \frac { 1 } { 2 } t ^ { 2 } } { \frac { 3 } { 6 } t + 2 } + C$$

Question 15

Solve the differential equation.
$$y ^ { \prime } = x e ^ { - \sin x } - y \cos x$$

Accepted Answer

A)  $y = \left( \frac { 1 } { 2 } x ^ { 2 } + C \right) e ^ { - \sin x }$ 
B) 
$y = \frac { 1 } { 2 } x + C e ^ { - \cos x }$ 
C)  $y = C e ^ { - \sin x }$ 
D)  $y = e ^ { - \sin x } + C x$ 
E)  $y = \left( 2 x ^ { 2 } + C \right) e ^ { - x } \sin x$ 
A)  $y = \left( \frac { 1 } { 2 } x ^ { 2 } + C \right) e ^ { - \sin x }$ 
B) 
$y = \frac { 1 } { 2 } x + C e ^ { - \cos x }$ 
C)  $y = C e ^ { - \sin x }$ 
D)  $y = e ^ { - \sin x } + C x$ 
E)  $y = \left( 2 x ^ { 2 } + C \right) e ^ { - x } \sin x$

Question 16

Suppose that a population develops according to the logistic equation
$$\frac { d P } { d t } = 0.02 P - 0.0002 P ^ { 2 }$$
where $t$ is measured in weeks. What is the carrying capacity?

Accepted Answer

A)  $K = 100$ 
B)  $K = 0.02$ 
C)  $K = 0.0002$ 
D)  $K = 300$ 
E)  $K = 200$ 
A)  $K = 100$ 
B)  $K = 0.02$ 
C)  $K = 0.0002$ 
D)  $K = 300$ 
E)  $K = 200$

Question 17

Solve the differential equation.
$$( 10 + t ) \frac { d u } { d t } + u = 10 + t , t > 0$$

Accepted Answer

The answer of Solve the differential equation.
\[( 10 + t...

Question 18

A certain small country has $\$ 20$ billion in paper currency in circulation, and each day $\$ 70$ million comes into the country's banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let $x = x ( t )$ denote the amount of new currency in circulation at time $t$ with $x ( 0 ) = 0$. Formulate and solve a mathematical model in the form of an initial-value problem that represents the "flow" of the new currency into circulation (in billions per day).

Accepted Answer

The answer of A certain small country has $\$ 20$...

Question 19

$$\text { Let } \frac { d P ( t ) } { d t } = 0.1 P \left( 1 - \frac { P } { 830 } \right) - \frac { 1,810 } { 83 } \text {. }$$ $$\text { What are the equilibrium solutions? }$$

Accepted Answer

The answer of \[\text { Let } \frac { d...

Question 20

Solve the differential equation.
$$4 \frac { d w } { d t } + 9 e ^ { t + w } = 0$$

Accepted Answer

A) 
$$w = \frac { 9 } { 4 } e ^ { t } - C$$ 
B) 
$$w = - \ln \left( \frac { 9 e ^ { t } } { 4 } - C \right)$$ 
C) 
$$w = \ln \left( \frac { 4 } { 9 } + \frac { 4 } { 9 } C e ^ { t } \right) - t$$ 
D) 
$$w = \ln \left( \frac { 4 } { 9 } t \right) - t$$ 
E)  none of these 
A) 
$$w = \frac { 9 } { 4 } e ^ { t } - C$$ 
B) 
$$w = - \ln \left( \frac { 9 e ^ { t } } { 4 } - C \right)$$ 
C) 
$$w = \ln \left( \frac { 4 } { 9 } + \frac { 4 } { 9 } C e ^ { t } \right) - t$$ 
D) 
$$w = \ln \left( \frac { 4 } { 9 } t \right) - t$$ 
E)  none of these

Solve the differential equation. $\frac { d u } { d t } = 15 + 5 u + 3 t + u t$

A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point $P$ is 3 times the y-coordinate $P$ . What is the equation of the curve?

Solve the initial-value problem. $\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 10$

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: $\frac { d A } { d t } = 2 A ( 1 - 0.0005 A ) - 0.01 A L$ $\frac { d L } { d t } = - 0.6 L + 0.0005 A L$ Find the equilibrium solution.

Suppose that a population grows according to a logistic model with carrying capacity 3,000 and $k = 0.02$ per year. Choose the logistic differential equation for these data.

Solve the differential equation. $( 8 + t ) \frac { d u } { d t } + u = 8 + t , t > 0$

Suppose that a population grows according to a logistic model with carrying capacity 7,300 and $k = 0.04$ per year. Write the logistic differential equation for these data.

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: $\frac { d A } { d t } = 2 A ( 1 - 0.0005 A ) - 0.01 A L$ $\frac { d L } { d t } = - 0.6 L + 0.0005 A L$ Find the equilibrium solution.

Solve the initial-value problem. $x y ^ { t } - y = x \ln x , \quad y ( 1 ) = 4$

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

Which equation does the function $y = e ^ { - 6 t }$ satisfy? Select the correct answer.

Determine whether the differential equation is linear. $y ^ { t } + 3 x ^ { 2 } y = 6 x ^ { 2 }$

Solve the differential equation. Select the correct answer. $( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0$

Solve the differential equation. $y ^ { \prime } = x e ^ { - \sin x } - y \cos x$

Suppose that a population develops according to the logistic equation $\frac { d P } { d t } = 0.02 P - 0.0002 P ^ { 2 }$ where $t$ is measured in weeks. What is the carrying capacity?

Solve the differential equation. $( 10 + t ) \frac { d u } { d t } + u = 10 + t , t > 0$

$\text { Let } \frac { d P ( t ) } { d t } = 0.1 P \left( 1 - \frac { P } { 830 } \right) - \frac { 1,810 } { 83 } \text {. }$ $\text { What are the equilibrium solutions? }$

Solve the differential equation. $4 \frac { d w } { d t } + 9 e ^ { t + w } = 0$

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Exam 9: Differential Equations

Solve the differential equation. dudt=15+5u+3t+ut\frac { d u } { d t } = 15 + 5 u + 3 t + u tdtdu​=15+5u+3t+ut

A curve passes through the point (8,2)( 8,2 )(8,2) and has the property that the slope of the curve at every point PPP is 3 times the y-coordinate PPP . What is the equation of the curve?

Solve the initial-value problem. drdt+2tr−r=0,r(0)=10\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 10dtdr​+2tr−r=0,r(0)=10

Suppose that a population grows according to a logistic model with carrying capacity 3,000 and k=0.02k = 0.02k=0.02 per year. Choose the logistic differential equation for these data.

Solve the differential equation. (8+t)dudt+u=8+t,t>0( 8 + t ) \frac { d u } { d t } + u = 8 + t , t > 0(8+t)dtdu​+u=8+t,t>0

Suppose that a population grows according to a logistic model with carrying capacity 7,300 and k=0.04k = 0.04k=0.04 per year. Write the logistic differential equation for these data.

Solve the initial-value problem. xyt−y=xln⁡x,y(1)=4x y ^ { t } - y = x \ln x , \quad y ( 1 ) = 4xyt−y=xlnx,y(1)=4

Solve the initial-value problem. drdt+2tr−r=0,r(0)=4\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 4dtdr​+2tr−r=0,r(0)=4

Which equation does the function y=e−6ty = e ^ { - 6 t }y=e−6t satisfy? Select the correct answer.

Determine whether the differential equation is linear. yt+3x2y=6x2y ^ { t } + 3 x ^ { 2 } y = 6 x ^ { 2 }yt+3x2y=6x2

Solve the differential equation. Select the correct answer. (6+t)dudt+u=6+t,t>0( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0(6+t)dtdu​+u=6+t,t>0

Solve the differential equation. y′=xe−sin⁡x−ycos⁡xy ^ { \prime } = x e ^ { - \sin x } - y \cos xy′=xe−sinx−ycosx

Suppose that a population develops according to the logistic equation dPdt=0.02P−0.0002P2\frac { d P } { d t } = 0.02 P - 0.0002 P ^ { 2 }dtdP​=0.02P−0.0002P2 where ttt is measured in weeks. What is the carrying capacity?

Solve the differential equation. (10+t)dudt+u=10+t,t>0( 10 + t ) \frac { d u } { d t } + u = 10 + t , t > 0(10+t)dtdu​+u=10+t,t>0

Solve the differential equation. 4dwdt+9et+w=04 \frac { d w } { d t } + 9 e ^ { t + w } = 04dtdw​+9et+w=0

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

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Solve the differential equation. $\frac { d u } { d t } = 15 + 5 u + 3 t + u t$

A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point $P$ is 3 times the y-coordinate $P$ . What is the equation of the curve?

Solve the initial-value problem. $\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 10$

Suppose that a population grows according to a logistic model with carrying capacity 3,000 and $k = 0.02$ per year. Choose the logistic differential equation for these data.

Solve the differential equation. $( 8 + t ) \frac { d u } { d t } + u = 8 + t , t > 0$

Suppose that a population grows according to a logistic model with carrying capacity 7,300 and $k = 0.04$ per year. Write the logistic differential equation for these data.

Solve the initial-value problem. $x y ^ { t } - y = x \ln x , \quad y ( 1 ) = 4$

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

Which equation does the function $y = e ^ { - 6 t }$ satisfy? Select the correct answer.

Determine whether the differential equation is linear. $y ^ { t } + 3 x ^ { 2 } y = 6 x ^ { 2 }$

Solve the differential equation. Select the correct answer. $( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0$

Solve the differential equation. $y ^ { \prime } = x e ^ { - \sin x } - y \cos x$

Suppose that a population develops according to the logistic equation $\frac { d P } { d t } = 0.02 P - 0.0002 P ^ { 2 }$ where $t$ is measured in weeks. What is the carrying capacity?

Solve the differential equation. $( 10 + t ) \frac { d u } { d t } + u = 10 + t , t > 0$

Solve the differential equation. $4 \frac { d w } { d t } + 9 e ^ { t + w } = 0$