Question 1

Suppose that a population grows according to a logistic model with carrying capacity 7,000 and $k = 0.05$ per year. Choose the logistic differential equation for these data.

Accepted Answer

A)  $\frac { d P ( t ) } { d t } = 7,000 P \left( 1 + \frac { p } { 0.05 } \right)$ 
B)  $\frac { d P ( t ) } { d t } = P \left( 1 + \frac { p } { 7,000 } \right)$ 
C)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 + \frac { p } { 7,000 } \right)$ 
D)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 + \frac { p } { 5 } \right)$ 
E)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 - \frac { p } { 7,000 } \right)$ 
A)  $\frac { d P ( t ) } { d t } = 7,000 P \left( 1 + \frac { p } { 0.05 } \right)$ 
B)  $\frac { d P ( t ) } { d t } = P \left( 1 + \frac { p } { 7,000 } \right)$ 
C)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 + \frac { p } { 7,000 } \right)$ 
D)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 + \frac { p } { 5 } \right)$ 
E)  $\frac { d P ( t ) } { d t } = 0.05 P \left( 1 - \frac { p } { 7,000 } \right)$

Question 2

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

Question 3

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

Accepted Answer

A)  $y = 11 x \sin x$ 
B)  $y = - x \sin x + 11 x$ 
C)  $y = - x \cos x - x$ 
D)  $y = 11 x \cos x - \sin x$ 
E)  $y = x \sin x - 11 x$ 
A)  $y = 11 x \sin x$ 
B)  $y = - x \sin x + 11 x$ 
C)  $y = - x \cos x - x$ 
D)  $y = 11 x \cos x - \sin x$ 
E)  $y = x \sin x - 11 x$

Question 4

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 5

A sum of $\$ 2170$ is invested at $11 \%$ 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 $\$ 2170$ is invested...

Question 6

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 7

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 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

The answer of Let \(\frac { d P ( t...

Question 10

Find the solution of the differential equation that satisfies the initial condition $y ( 0 ) = 1$.
$$\frac { d y } { d x } = 6 x ^ { 5 } y$$

Accepted Answer

The answer of Find the solution of the differential equation...

Question 11

Solve the initial-value problem. $x y ^ { t } - y = x \ln x , y ( 1 ) = 3$

Accepted Answer

A)  $y ( x ) = 3 x ( \ln x ) ^ { 2 }$ 
B)  $y ( x ) = \frac { 1 } { 2 } x ( \ln x ) ^ { 2 } + 3 x$ 
C) $y ( x ) = 3 x \ln x + \frac { 1 } { 2 } x$ 
D) $y ( x ) = 3 \ln x + \frac { 1 } { 2 } x$ 
E)  $y ( x ) = \frac { 1 } { 2 } ( \ln x ) ^ { 3 } + x$ 
A)  $y ( x ) = 3 x ( \ln x ) ^ { 2 }$ 
B)  $y ( x ) = \frac { 1 } { 2 } x ( \ln x ) ^ { 2 } + 3 x$ 
C) $y ( x ) = 3 x \ln x + \frac { 1 } { 2 } x$ 
D) $y ( x ) = 3 \ln x + \frac { 1 } { 2 } x$ 
E)  $y ( x ) = \frac { 1 } { 2 } ( \ln x ) ^ { 3 } + x$

Question 12

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

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

Question 13

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

Accepted Answer

The answer of Kirchhoff's Law gives us the derivative equation...

Question 14

A sum of $\$ 2,000$ is invested at $20 \%$ 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

A)  $\frac { d A } { d t } = 20 A , \quad A ( 0 ) = 3,000$ 
B)  $\frac { d A } { d t } = 1,500 A , \quad A ( 0 ) = 2,000$ 
C)  $\frac { d A } { d t } = 0.20 A , A ( 0 ) = 200$ 
D)  $\frac { d A } { d t } = 0.20 A ( 0 ) , \quad A ( 0 ) = 2,000$ 
E)  $\frac { d A } { d t } = 0.20 A , \quad A ( 0 ) = 2,000$ 
A)  $\frac { d A } { d t } = 20 A , \quad A ( 0 ) = 3,000$ 
B)  $\frac { d A } { d t } = 1,500 A , \quad A ( 0 ) = 2,000$ 
C)  $\frac { d A } { d t } = 0.20 A , A ( 0 ) = 200$ 
D)  $\frac { d A } { d t } = 0.20 A ( 0 ) , \quad A ( 0 ) = 2,000$ 
E)  $\frac { d A } { d t } = 0.20 A , \quad A ( 0 ) = 2,000$

Question 15

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 16

Solve the differential equation.
$4 \frac { d w } { d t } + 5 e ^ { t + w } = 0$

Accepted Answer

A) $w = \frac { 5 } { 4 } e ^ { t } - C$ 
B) $w = \ln \left( \frac { 4 } { 5 } t \right) - t$ 
C) 
$w = - \ln \left( \frac { 5 e ^ { t } } { 4 } - C \right)$ 
D) 
$w = \ln \left( \frac { 4 } { 5 } + \frac { 4 } { 5 } C e ^ { t } \right) - t$ 
E)  none of these 
A) $w = \frac { 5 } { 4 } e ^ { t } - C$ 
B) $w = \ln \left( \frac { 4 } { 5 } t \right) - t$ 
C) 
$w = - \ln \left( \frac { 5 e ^ { t } } { 4 } - C \right)$ 
D) 
$w = \ln \left( \frac { 4 } { 5 } + \frac { 4 } { 5 } C e ^ { t } \right) - t$ 
E)  none of these

Question 17

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 10,700 . The number of fish tripled in the first year. Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after $t$ years.

Accepted Answer

The answer of Biologists stocked a lake with 400 fish...

Question 18

$y = C e ^ { 4 x ^ { 2 } }$ is the solution of the differential equation $y ^ { \prime } = 8 x y$. Find the solution that satisfies the initial condition $y ( 1 ) = 1$.

Accepted Answer

The answer of \(y = C e ^ { 4...

Question 19

Use Euler's method with step size $0.25$ to estimate $y ( 1 )$, where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places.
$$y ^ { t } = 4 x + y ^ { 2 } , y ( 0 ) = 0$$

Accepted Answer

A)  $y ( 1 ) = 4.1622$ 
B)  $y ( 1 ) = 1.6622$ 
C)  $y ( 1 ) = 3.1622$ 
D)  $y ( 1 ) = 3.1622$. 
E)  $y ( 1 ) = 2.6622$. 
A)  $y ( 1 ) = 4.1622$ 
B)  $y ( 1 ) = 1.6622$ 
C)  $y ( 1 ) = 3.1622$ 
D)  $y ( 1 ) = 3.1622$. 
E)  $y ( 1 ) = 2.6622$.

Question 20

Solve the differential equation. $$y ^ { t } = x ^ { 5 } e ^ { - \sin x } - y \cos x$$

Accepted Answer

A)  $y = \left( 6 x ^ { 6 } + C \right) e ^ { - x } \sin x$ 
B)  $y = 5 \mathrm { Ce } e ^ { - \sin x }$ 
C)  $y = \left( \frac { 1 } { 6 } x ^ { 6 } + C \right) e ^ { - \sin x }$ 
D)  $y = e ^ { - \sin x } + 5 C x ^ { 5 }$ 
E)  $y = \frac { 1 } { 6 } x + C e ^ { - \cos x }$ 
A)  $y = \left( 6 x ^ { 6 } + C \right) e ^ { - x } \sin x$ 
B)  $y = 5 \mathrm { Ce } e ^ { - \sin x }$ 
C)  $y = \left( \frac { 1 } { 6 } x ^ { 6 } + C \right) e ^ { - \sin x }$ 
D)  $y = e ^ { - \sin x } + 5 C x ^ { 5 }$ 
E)  $y = \frac { 1 } { 6 } x + C e ^ { - \cos x }$

Suppose that a population grows according to a logistic model with carrying capacity 7,000 and $k = 0.05$ per year. Choose the logistic differential equation for these data.

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

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

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.

A sum of $\$ 2170$ is invested at $11 \%$ 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 $A ( t )$ .

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. $x y ^ { t } = y + x ^ { 2 } \cos x , y ( \pi ) = 0$

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

Find the solution of the differential equation that satisfies the initial condition $y ( 0 ) = 1$ . $\frac { d y } { d x } = 6 x ^ { 5 } y$

Solve the initial-value problem. $x y ^ { t } - y = x \ln x , y ( 1 ) = 3$

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

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

A sum of $\$ 2,000$ is invested at $20 \%$ 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 differential equation. $\frac { d u } { d t } = 15 + 5 u + 3 t + u t$

Solve the differential equation. $4 \frac { d w } { d t } + 5 e ^ { t + w } = 0$

$y = C e ^ { 4 x ^ { 2 } }$ is the solution of the differential equation $y ^ { \prime } = 8 x y$ . Find the solution that satisfies the initial condition $y ( 1 ) = 1$ .

Use Euler's method with step size $0.25$ to estimate $y ( 1 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { t } = 4 x + y ^ { 2 } , y ( 0 ) = 0$

Solve the differential equation. $y ^ { t } = x ^ { 5 } e ^ { - \sin x } - y \cos x$

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Exam 9: Differential Equations

Suppose that a population grows according to a logistic model with carrying capacity 7,000 and k=0.05k = 0.05k=0.05 per year. Choose the logistic differential equation for these data.

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

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 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.

A sum of $2170\$ 2170$2170 is invested at 11%11 \%11% 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) .

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. xyt=y+x2cos⁡x,y(π)=0x y ^ { t } = y + x ^ { 2 } \cos x , y ( \pi ) = 0xyt=y+x2cosx,y(π)=0

Let dP(t)dt=0.1P(1−P830)−1,81083\frac { d P ( t ) } { d t } = 0.1 P \left( 1 - \frac { P } { 830 } \right) - \frac { 1,810 } { 83 }dtdP(t)​=0.1P(1−830P​)−831,810​ .What are the equilibrium solutions?

Find the solution of the differential equation that satisfies the initial condition y(0)=1y ( 0 ) = 1y(0)=1 . dydx=6x5y\frac { d y } { d x } = 6 x ^ { 5 } ydxdy​=6x5y

Solve the initial-value problem. xyt−y=xln⁡x,y(1)=3x y ^ { t } - y = x \ln x , y ( 1 ) = 3xyt−y=xlnx,y(1)=3

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

Kirchhoff's Law gives us the derivative equation Qt=12−4QQ ^ { t } = 12 - 4 QQt=12−4Q . If Q(0)=0Q ( 0 ) = 0Q(0)=0 , use Euler's method with step size 0.10.10.1 to estimate QQQ after 0.30.30.3 second.

A sum of $2,000\$ 2,000$2,000 is invested at 20%20 \%20% 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 differential equation. dudt=15+5u+3t+ut\frac { d u } { d t } = 15 + 5 u + 3 t + u tdtdu​=15+5u+3t+ut

Solve the differential equation. 4dwdt+5et+w=04 \frac { d w } { d t } + 5 e ^ { t + w } = 04dtdw​+5et+w=0

y=Ce4x2y = C e ^ { 4 x ^ { 2 } }y=Ce4x2 is the solution of the differential equation y′=8xyy ^ { \prime } = 8 x yy′=8xy . Find the solution that satisfies the initial condition y(1)=1y ( 1 ) = 1y(1)=1 .

Use Euler's method with step size 0.250.250.25 to estimate y(1)y ( 1 )y(1) , where y(x)y ( x )y(x) is the solution of the initial-value problem. Round your answer to four decimal places. yt=4x+y2,y(0)=0y ^ { t } = 4 x + y ^ { 2 } , y ( 0 ) = 0yt=4x+y2,y(0)=0

Solve the differential equation. yt=x5e−sin⁡x−ycos⁡xy ^ { t } = x ^ { 5 } e ^ { - \sin x } - y \cos xyt=x5e−sinx−ycosx

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Suppose that a population grows according to a logistic model with carrying capacity 7,000 and $k = 0.05$ per year. Choose the logistic differential equation for these data.

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

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

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.

A sum of $\$ 2170$ is invested at $11 \%$ 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 $A ( t )$ .

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. $x y ^ { t } = y + x ^ { 2 } \cos x , y ( \pi ) = 0$

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

Find the solution of the differential equation that satisfies the initial condition $y ( 0 ) = 1$ . $\frac { d y } { d x } = 6 x ^ { 5 } y$

Solve the initial-value problem. $x y ^ { t } - y = x \ln x , y ( 1 ) = 3$

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

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

A sum of $\$ 2,000$ is invested at $20 \%$ 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 differential equation. $\frac { d u } { d t } = 15 + 5 u + 3 t + u t$

Solve the differential equation. $4 \frac { d w } { d t } + 5 e ^ { t + w } = 0$

$y = C e ^ { 4 x ^ { 2 } }$ is the solution of the differential equation $y ^ { \prime } = 8 x y$ . Find the solution that satisfies the initial condition $y ( 1 ) = 1$ .

Use Euler's method with step size $0.25$ to estimate $y ( 1 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { t } = 4 x + y ^ { 2 } , y ( 0 ) = 0$

Solve the differential equation. $y ^ { t } = x ^ { 5 } e ^ { - \sin x } - y \cos x$