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Exam 9: Differential Equations

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Question 71

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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) 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 .

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Question 72

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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 $35 \mathrm {~min}$ ? Select the correct answer.

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

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Question 73

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Select a direction field for the differential equation $y ^ { \prime } = y ^ { 2 } - x ^ { 2 }$ from a set of direction fields labeled I-IV.

Consider a population $P = P ( t )$ with constant relative birth and death rates $a$ and $\beta$ , respectively, and a constant emigration rate $m$ , where $\alpha = 0.9 , \beta = 0.7$ and $m = 0.8$ . Then the rate of change of the population at time $t$ is modeled by the differential equation $\frac { d P } { d t } = k P - m$ where $k = \alpha - \beta$ Find the solution of this equation with the rate of change of the population at time $t = 6$ that satisfies the initial condition $P ( 0 ) = 2200$ .

Suppose that a population develops according to the logistic equation $\frac { d P } { d t } = 0.05 P - 0.0005 P ^ { 2 }$ where $t$ is measured in weeks. What is the carrying capacity?

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 $35 \mathrm {~min}$ ?

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 $k y ^ { 102 }$ . If 7 such rabbits breed initially and the warren has 21 rabbits after 8 months, then when is doomsday?

Solve the differential equation. $y ^ { t } = x ^ { 4 } e ^ { - \sin x } - y \cos x$

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

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

Determine whether the differential equation is linear. Select the correct answer. $y ^ { t } + 7 x ^ { 6 } y = 6 x ^ { 6 }$

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

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 $\frac { d P } { d t } = r ( D - P ( t ) )$ as a reasonable model for learning, where $r$ is a positive constant. Solve it as a linear differential equation.

$\text { Which equation does the function } y = e ^ { - 6 t } \text { satisfy? }$

Solve the initial-value problem. $r ^ { t } + 2 t r = r , r ( 0 ) = 5$

Determine whether the differential equation is linear. Select the correct answer. $y ^ { t } + 6 x ^ { 5 } y = 6 x ^ { 5 }$

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 2,000 inhabitants, 130 people have a disease at the beginning of the week and 1,100 have it at the end of the week. How long does it take for $80 \%$ of the population to be infected?

Suppose that a population develops according to the logistic equation $\frac { d P } { d t } = 0.04 P - 0.0004 P ^ { 2 }$ where $t$ is measured in weeks. What is the carrying capacity? Select the correct answer.

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

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