Question 1

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)  $k = \frac { 9 } { 7 }$ C, E

Question 2

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

Accepted Answer

$$u = - 11 + C e ^ { \frac { t ^ { 2 } } { 2 } + 2 t }$$

Question 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 } = 2 A ( 1 - 0.0005 A ) - 0.01 A L\) $\frac { d L } { d t } = - 0.6 L + 0.0005 \mathrm { AL }$
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)  $A = 1500 , L = 100$ D

Question 4

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 5

Suppose that a population grows according to a logistic model with carrying capacity 2,000 and $k = 0.05$ per year. Choose the logistic differential equation for these data.

Accepted Answer

A)  $\frac { d P ( t ) } { d t } = P \left( 1 + \frac { p } { 2,000 } \right)$ 
B)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 + \frac { p } { 5 } \right)$ 
C)  $\frac { d P ( t ) } { d t } = 2,000 P \left( 1 + \frac { p } { 0.05 } \right)$ 
D)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 + \frac { p } { 2,000 } \right)$ 
E)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 - \frac { p } { 2,000 } \right)$ 
A)  $\frac { d P ( t ) } { d t } = P \left( 1 + \frac { p } { 2,000 } \right)$ 
B)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 + \frac { p } { 5 } \right)$ 
C)  $\frac { d P ( t ) } { d t } = 2,000 P \left( 1 + \frac { p } { 0.05 } \right)$ 
D)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 + \frac { p } { 2,000 } \right)$ 
E)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 - \frac { p } { 2,000 } \right)$

Question 6

$\text { A function } y ( t ) \text { satisfies the differential equation } \frac { d y } { d t } = y ^ { 4 } - 9 y ^ { 3 } + 20 y ^ { 2 } \text {. }$ What are the constant solutions of the equation?

Accepted Answer

The answer of \(\text { A function } y (...

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 50 cells. Find the number of cells after 6 hours.

Accepted Answer

A)  $P = 13,119,200$ 
B)  $P = 13,107,200$ 
C)  $P = 13,227,200$ 
D)  $P = 13,207,200$ 
E)  $P = 13,247,200$ 
A)  $P = 13,119,200$ 
B)  $P = 13,107,200$ 
C)  $P = 13,227,200$ 
D)  $P = 13,207,200$ 
E)  $P = 13,247,200$

Question 8

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 9

Which of the following functions are the constant solutions of the equation $$\frac { d y } { d t } = y ^ { 4 } - y ^ { 3 } + 6 y ^ { 2 }$$
a. $y ( t ) = 2$
b. $y ( t ) = 3$
c. $y ( t ) = 5$
d. $y ( t ) = 0$
e. $y ( t ) = e ^ { t }$

Accepted Answer

The answer of Which of the following functions are the...

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

Solve the initial-value problem. $r ^ { t } + 2 t r = r , r ( 0 ) = 6$

Accepted Answer

A) 
$r ( t ) = \frac { e ^ { t ^ { 2 } } } { 6 }$ 
B)  $r ( t ) = 6 e ^ { t - t ^ { 2 } }$ 
C)  $r ( t ) = e ^ { t - 6 t ^ { 2 } }$ 
D)  $r ( t ) = 6 e ^ { t - 2 t }$ 
E)  $r ( t ) = 6 e ^ { 2 t }$ 
A) 
$r ( t ) = \frac { e ^ { t ^ { 2 } } } { 6 }$ 
B)  $r ( t ) = 6 e ^ { t - t ^ { 2 } }$ 
C)  $r ( t ) = e ^ { t - 6 t ^ { 2 } }$ 
D)  $r ( t ) = 6 e ^ { t - 2 t }$ 
E)  $r ( t ) = 6 e ^ { 2 t }$

Question 12

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 $\frac { d u } { d t } = k ( u - 60 ) \text {. }$ 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 13

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 14

Find the orthogonal trajectories of the family of curves. $$y = ( x + k ) ^ { - 8 }$$

Accepted Answer

The answer of Find the orthogonal trajectories of the family...

Question 15

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

Accepted Answer

A)  $y = C e ^ { - 2 / x }$ 
B)  $y = x ^ { 2 } e ^ { - 1 / x } + C e ^ { - 1 / x }$ 
C)  $y = C x e ^ { 2 x }$ 
D)  $y = e ^ { 1 / x } x + C e ^ { 1 / x }$ 
E)  none of these 
A)  $y = C e ^ { - 2 / x }$ 
B)  $y = x ^ { 2 } e ^ { - 1 / x } + C e ^ { - 1 / x }$ 
C)  $y = C x e ^ { 2 x }$ 
D)  $y = e ^ { 1 / x } x + C e ^ { 1 / x }$ 
E)  none of these

Question 16

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 17

$\text { Find the solution of the initial-value problem and use it to find the population when } t = 10 \text {. }$ $\frac { d P } { d t } = 0.4 P \left( 1 - \frac { P } { 2000 } \right) , P ( 0 ) = 100$

Accepted Answer

The answer of \(\text { Find the solution of the...

Question 18

A curve passes through the point $( 6,2 )$ and has the property that the slope of the curve at every point $P$ is 4 times the y-coordinate $P$. What is the equation of the curve?

Accepted Answer

A)  $y = 2 e ^ { 4 x - 6 }$ 
B)  $y = 2 e ^ { x - 24 }$ 
C)  $y = 2 e ^ { 4 x + 24 }$ 
D) $y = \frac { e ^ { 4 x - 24 } } { 2 }$ 
E)  $y = 2 e ^ { 4 x - 24 }$ 
A)  $y = 2 e ^ { 4 x - 6 }$ 
B)  $y = 2 e ^ { x - 24 }$ 
C)  $y = 2 e ^ { 4 x + 24 }$ 
D) $y = \frac { e ^ { 4 x - 24 } } { 2 }$ 
E)  $y = 2 e ^ { 4 x - 24 }$

Question 19

Let $c$ be a positive number. A differential equation of the form $\frac { d y } { d t } = k y ^ { 1 + c }$ where $k$ is a positive constant is called a doomsday equation because the exponent in the expression $k y ^ { 1 + c }$ is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term $k y ^ { 100 }$. If 3 such rabbits breed initially and the warren has 27 rabbits after 3 months, then when is doomsday?

Accepted Answer

A)  $47.03$ months 
B)  $97.03$ months 
C)  $67.03$ months 
D)  $137.03$ months 
E)  $117.03$ months 
A)  $47.03$ months 
B)  $97.03$ months 
C)  $67.03$ months 
D)  $137.03$ months 
E)  $117.03$ months

Question 20

Solve the differential equation. $3 y y ^ { t } = 7 x$

Accepted Answer

A)  $7 x ^ { 2 } + 3 y ^ { 2 } = C$ 
B)  $3 x ^ { 2 } + 7 y ^ { 2 } = C$ 
C)  $3 x ^ { 2 } - 7 y ^ { 2 } = C$ 
D)  $7 x ^ { 2 } - 3 y ^ { 2 } = C$ 
E)  $7 x ^ { 2 } + 3 y ^ { 2 } = 10$ 
A)  $7 x ^ { 2 } + 3 y ^ { 2 } = C$ 
B)  $3 x ^ { 2 } + 7 y ^ { 2 } = C$ 
C)  $3 x ^ { 2 } - 7 y ^ { 2 } = C$ 
D)  $7 x ^ { 2 } - 3 y ^ { 2 } = C$ 
E)  $7 x ^ { 2 } + 3 y ^ { 2 } = 10$

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. $\frac { d u } { d t } = 22 + 2 u + 11 t + u t$

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\) $\frac { d L } { d t } = - 0.6 L + 0.0005 \mathrm { AL }$ Find the equilibrium solution.

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

Suppose that a population grows according to a logistic model with carrying capacity 2,000 and $k = 0.05$ per year. Choose the logistic differential equation for these data.

$\text { A function } y ( t ) \text { satisfies the differential equation } \frac { d y } { d t } = y ^ { 4 } - 9 y ^ { 3 } + 20 y ^ { 2 } \text {. }$ What are the constant solutions of the equation?

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 50 cells. Find the number of cells after 6 hours.

Which of the following functions are the constant solutions of the equation $\frac { d y } { d t } = y ^ { 4 } - y ^ { 3 } + 6 y ^ { 2 }$ a. $y ( t ) = 2$ b. $y ( t ) = 3$ c. $y ( t ) = 5$ d. $y ( t ) = 0$ e. $y ( t ) = e ^ { t }$

Solve the initial-value problem. $r ^ { t } + 2 t r = r , r ( 0 ) = 6$

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?

Find the orthogonal trajectories of the family of curves. $y = ( x + k ) ^ { - 8 }$

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

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?

$\text { Find the solution of the initial-value problem and use it to find the population when } t = 10 \text {. }$ $\frac { d P } { d t } = 0.4 P \left( 1 - \frac { P } { 2000 } \right) , P ( 0 ) = 100$

A curve passes through the point $( 6,2 )$ and has the property that the slope of the curve at every point $P$ is 4 times the y-coordinate $P$ . What is the equation of the curve?

Solve the differential equation. $3 y y ^ { t } = 7 x$

Functions and Models

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Exam 9: Differential Equations

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

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

Suppose that a population grows according to a logistic model with carrying capacity 2,000 and k=0.05k = 0.05k=0.05 per year. Choose the logistic differential equation for these data.

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 50 cells. Find the number of cells after 6 hours.

Solve the initial-value problem. rt+2tr=r,r(0)=6r ^ { t } + 2 t r = r , r ( 0 ) = 6rt+2tr=r,r(0)=6

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?

Find the orthogonal trajectories of the family of curves. y=(x+k)−8y = ( x + k ) ^ { - 8 }y=(x+k)−8

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

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?

A curve passes through the point (6,2)( 6,2 )(6,2) and has the property that the slope of the curve at every point PPP is 4 times the y-coordinate PPP . What is the equation of the curve?

Solve the differential equation. 3yyt=7x3 y y ^ { t } = 7 x3yyt=7x

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

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. $\frac { d u } { d t } = 22 + 2 u + 11 t + u t$

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

Suppose that a population grows according to a logistic model with carrying capacity 2,000 and $k = 0.05$ per year. Choose the logistic differential equation for these data.

Solve the initial-value problem. $r ^ { t } + 2 t r = r , r ( 0 ) = 6$

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?

Find the orthogonal trajectories of the family of curves. $y = ( x + k ) ^ { - 8 }$

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

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?

A curve passes through the point $( 6,2 )$ and has the property that the slope of the curve at every point $P$ is 4 times the y-coordinate $P$ . What is the equation of the curve?

Solve the differential equation. $3 y y ^ { t } = 7 x$