Question 1

Suppose that a population grows according to a logistic model with carrying capacity  and  per year. Choose the logistic differential equation for these data.

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Question 2

Let  .
What are the equilibrium solutions?

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What are the equilibrium solutions?...

Question 3

The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s 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.)

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The answer of The population of the world was about...

Question 4

The rate of change of atmospheric pressure P with respect to altitude h is proportional to P provided that the temperature is constant. At  the pressure is  at sea level and  at  . What is the pressure at an altitude of  ?

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Question 5

Let c be a positive number. A differential equation of the form  where k is a positive constant, is called a doomsday equation because the exponent in the expression  is larger than the exponent 1for natural growth. An especially prolific breed of rabbits has the growth term  . If  such rabbits breed initially and the warren has  rabbits after  months, then when is doomsday?

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Question 6

Consider the differential equation  as a model for a fish population, where t is measured in weeks and c is a constant. For what values of c does the fish population always die out?

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Question 7

One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction of the population who have heard the rumor and the fraction who have not heard the rumor. Let's assume that the constant of proportionality is  . Write a differential equation that is satisfied by y.

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Question 8

The Pacific halibut fishery has been modeled by the differential equation  where  is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be  and  per year. If  , find the biomass a year later.

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Question 9

Suppose that a population grows according to a logistic model with carrying capacity  and  per year. Write the logistic differential equation for these data.

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Question 10

Let c be a positive number. A differential equation of the form  where k is a positive constant is called a doomsday equation because the exponent in the expression  is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term  . If  such rabbits breed initially and the warren has  rabbits after  months, then when is doomsday?

Accepted Answer

The given growth term for rabbits is $y'=ky^2$, where $y$ is the number of rabbits and $k$ is a positive constant. This is of the form given in the question: $y' = k\cdot y^{\frac{11}{5}}$.

We are given that there are $y(0) = 11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11$ rabbits initially and $y(\Delta t) = 11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11$ rabbits after $\Delta t = 11eaa3e5_5528_fff1_9f8f_cf2a8d302a89_TB5971_11$ months.

Using separation of variables, we have:

$$\frac{dy}{dt} = ky^{\frac{11}{5}}$$

$$\int \frac{dy}{y^{\frac{11}{5}}} = \int k\ dt$$

$$-\frac{5}{6} y^{-\frac{6}{5}} = kt+C$$

Using the initial condition $y(0) = 11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11$:

$$-\frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}} = k\cdot 0+C$$

Simplifying this expression, we get:

$$C = -\frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Plugging this value of $C$ back into our equation:

$$-\frac{5}{6} y^{-\frac{6}{5}} = kt-\frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Using the second initial condition $y(\Delta t) = 11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11$:

$$-\frac{5}{6} (11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11)^{-\frac{6}{5}} = k\cdot \Delta t-\frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

We want to find the value of $\Delta t$ for which $y$ becomes infinite (i.e. doomsday), so we set $y$ to infinity:

$$-\frac{5}{6} \cdot \infty^{-\frac{6}{5}} = k\cdot t - \frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Simplifying this expression, we get:

$$-\frac{5}{6} \cdot 0 = k\cdot \Delta t - \frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

$$k\cdot \Delta t = \frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Plugging in the given values, we get:

$$k\cdot \Delta t = \frac{5}{6} (11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11)^{-\frac{6}{5}} - \frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Simplifying this expression using a common denominator, we get:

$$k\cdot \Delta t = \frac{5}{6} \cdot \frac{(11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11)^{-\frac{6}{5}} - (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}}{(11eaa3e5_5528_fff1_9f8f_cf2a8d302a89_TB5971_11) - 0}$$

Using a calculator or Wolfram Alpha, we can find that the value of $\Delta t$ that satisfies this equation is approximately $11eaa3e5_5529_2705_9f8f_b9c563b683c4_TB5971_11$, so the answer is $\boxed{	extbf{(D)}}$.

Question 11

A curve passes through the point  and has the property that the slope of the curve at every point P is  times the y-coordinate P. What is the equation of the curve?

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Question 12

Biologists stocked a lake with  fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be  . 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.

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Question 13

Solve the differential equation.

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The answer of Solve the differential equation. 
...

Question 14

The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s 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

The answer of The population of the world was about...

Question 15

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  inhabitants,  people have a disease at the beginning of the week and  have it at the end of the week. How long does it take for  of the population to be infected?

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Question 16

A sum of  is invested at  interest. If  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  .

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Question 17

Suppose that a population develops according to the logistic equation  , where t is measured in weeks. What is the carrying capacity?

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Question 18

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  inhabitants,  people have a disease at the beginning of the week and  have it at the end of the week. How long does it take for  of the population to be infected?

Accepted Answer

The answer of One model for the spread of an...

Question 19

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  . The initial population of a culture is  cells. Find the number of cells after  hours.

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

Question 20

Solve the differential equation.

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The answer of Solve the differential equation. 
...

Quiz 9: Differential Equations

Suppose that a population grows according to a logistic model with carrying capacity and per year. Choose the logistic differential equation for these data.

Let . What are the equilibrium solutions?

The rate of change of atmospheric pressure P with respect to altitude h is proportional to P provided that the temperature is constant. At the pressure is at sea level and at . What is the pressure at an altitude of ?

Consider the differential equation as a model for a fish population, where t is measured in weeks and c is a constant. For what values of c does the fish population always die out?

The Pacific halibut fishery has been modeled by the differential equation where is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be and per year. If , find the biomass a year later.

Suppose that a population grows according to a logistic model with carrying capacity and per year. Write the logistic differential equation for these data.

A curve passes through the point and has the property that the slope of the curve at every point P is times the y-coordinate P. What is the equation of the curve?

Solve the differential equation.

A sum of is invested at interest. If 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 .

Suppose that a population develops according to the logistic equation , where t is measured in weeks. What is the carrying capacity?

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 . The initial population of a culture is cells. Find the number of cells after hours.

Solve the differential equation.