Chapter 9

Give the rates of growth of two populations, \(x\) and \(y,\) measured in thousands. (a) Describe in words what happens to the population of each species in the absence of the other. (b) Describe in words how the species interact with one another. Give reasons why the populations might behave as described by the equations. Suggest species that might interact in that way. $$\begin{aligned} &\frac{d x}{d t}=0.2 x\\\ &\frac{d y}{d t}=0.4 x y-0.1 y \end{aligned}$$

Let \(I\) be the number of infected people and \(S\) be the number of susceptible people in an outbreak of a disease. Explain why it is reasonable to model the interaction between these two groups by the differential equations \(\frac{d S}{d t}=-a S I\) \(\frac{d I}{d t}=a S I-b I \quad\) where \(a, b\) are positive constants. Why have the signs been chosen this way? Why is the constant \(a\) the same in both equations?

Find particular solutions. $$\frac{d y}{d t}=0.5(y-200), \quad y=50\( when \)t=0$$

Find solutions to the differential equations in subject to the given initial condition. $$\frac{d P}{d t}=0.02 P, \quad P(0)=20$$

Decide whether or not each of the following is a solution to the differential equation \(x y^{\prime}-2 y=0\) (a) \(y=x^{2}\) (b) \(y=x^{3}\)

Give the rates of growth of two populations, \(x\) and \(y,\) measured in thousands. (a) Describe in words what happens to the population of each species in the absence of the other. (b) Describe in words how the species interact with one another. Give reasons why the populations might behave as described by the equations. Suggest species that might interact in that way. $$\begin{aligned} &\frac{d x}{d t}=0.01 x-0.05 x y\\\ &\frac{d y}{d t}=-0.2 y+0.08 x y \end{aligned}$$

Find particular solutions. $$\frac{d P}{d t}=P+4, \quad P=100\( when \)t=0$$

Find solutions to the differential equations in subject to the given initial condition. $$\frac{d w}{d r}=3 w, \quad w=30 \text { when } r=0$$

Check that \(y=t^{4}\) is a solution to the differential equation \(t \frac{d y}{d t}=4 y\)

Give the rates of growth of two populations, \(x\) and \(y,\) measured in thousands. (a) Describe in words what happens to the population of each species in the absence of the other. (b) Describe in words how the species interact with one another. Give reasons why the populations might behave as described by the equations. Suggest species that might interact in that way. $$\begin{aligned} &\frac{d x}{d t}=0.01 x-0.05 x y\\\ &\frac{d y}{d t}=0.2 y-0.08 x y \end{aligned}$$

Find particular solutions. $$\frac{d H}{d t}=3(H-75), \quad H=0\( when \)t=0$$

A population of insects grows at a rate proportional to the size of the population. Write a differential equation for the size of the population, \(P\), as a function of time, \(t\) Is the constant of proportionality positive or negative?

Find solutions to the differential equations in subject to the given initial condition. $$\frac{d y}{d x}=-0.14 y, \quad y=5.6 \text { when } x=0$$

Find the general solution to the differential equation $$\frac{d y}{d t}=2 t$$

The following system of differential equations represents the interaction between two populations, \(x\) and \(y\) $$\begin{aligned} &\frac{d x}{d t}=-3 x+2 x y\\\ &\frac{d y}{d t}=-y+5 x y \end{aligned}$$ (a) Describe how the species interact. How would each species do in the absence of the other? Are they helpful or harmful to each other? (b) If \(x=2\) and \(y=1,\) does \(x\) increase or decrease? Does \(y\) increase or decrease? Justify your answers. (c) Write a differential equation involving \(d y / d x\) (d) Use a computer or calculator to draw the slope field for the differential equation in part (c). (e) Draw the trajectory starting at point \(x=2, y=1\) on your slope field, and describe how the populations change as time increases.

(a) In a school of 150 students, one of the students has the flu initially. What is \(I_{0} ?\) What is \(S_{0} ?\) (b) Use these values of \(I_{0}\) and \(S_{0}\) and the equation $$\frac{d I}{d t}=0.0026 S I-0.5 I$$ to determine whether the number of infected people initially increases or decreases. What does this tell you about the spread of the disease?

Find particular solutions. $$\frac{d m}{d t}=0.1 m+200, \quad m(0)=1000$$

Money in a bank account earns interest at a continuous annual rate of \(5 \%\) times the current balance. Write a differential equation for the balance, \(B\), in the account as a function of time, \(t,\) in years.

Find solutions to the differential equations in subject to the given initial condition. $$\frac{d Q}{d t}=\frac{Q}{5}, \quad Q=50 \text { when } t=0$$

Create a system of differential equations to model the situations. You may assume that all constants of proportionality are 1. The concentrations of two chemicals are denoted by \(x\) and \(y,\) respectively. Alone, each decays at a rate proportional to its concentration. Together, they interact to form a third substance. As the third substance is created, the concentrations of the initial two populations get smaller.

Find particular solutions. $$\frac{d B}{d t}=4 B-100, \quad B=20\( when \)t=0$$

Radioactive substances decay at a rate proportional to the quantity present. Write a differential equation for the quantity, \(Q,\) of a radioactive substance present at time \(t\) Is the constant of proportionality positive or negative?

Find solutions to the differential equations in subject to the given initial condition. $$\frac{d p}{d q}=-0.1 p, \quad p=100 \text { when } q=5$$

Create a system of differential equations to model the situations. You may assume that all constants of proportionality are 1. Two businesses are in competition with each other. Both businesses would do well without the other one, but each hurts the other's business. The values of the two businesses are given by \(x\) and \(y\).

Find particular solutions. $$\frac{d Q}{d t}=0.3 Q-120, \quad Q=50 \text { when } t=0$$

(a) For \(d y / d x=x^{2}-y^{2},\) find the slope at the following points: \((1,0), \quad(0,1), \quad(1,1)\) \((2,1), \quad(1,2), \quad 2\) (2,2) (b) Sketch the slope ficld at these points.

A bank account that initially contains $$ 25,000\( earns interest at a continuous rate of \)4 \%\( per year. Withdrawals are made out of the account at a constant rate of $$ 2000 per year. Write a differential equation for the balance, \)B\( in the account as a function of the number of years, \)t$

Find solutions to the differential equations in subject to the given initial condition. $$\frac{d y}{d x}+\frac{y}{3}=0, \quad y(0)=10$$

Create a system of differential equations to model the situations. You may assume that all constants of proportionality are 1. A population of fleas is represented by \(x,\) and a population of dogs is represented by \(y .\) The fleas need the dogs in order to survive. The dog population, however, is unaffected by the fleas.

Find particular solutions. $$\frac{d B}{d t}+2 B=50, \quad B(1)=100$$

A pollutant spilled on the ground decays at a rate of \(8 \%\) a day. In addition, cleanup crews remove the pollutant at a rate of 30 gallons a day. Write a differential equation for the amount of pollutant, \(P\), in gallons, left after \(t\) days.

A deposit of \(\$ 5000\) is made to a bank account paying \(1.5 \%\) annual interest, compounded continuously. (a) Write a differential equation for the balance in the account, \(B\), as a function of time, \(t\), in years. (b) Solve the differential equation. (c) How much money is in the account in 10 years?

Find particular solutions. $$\frac{d B}{d t}+0.1 B-10=0 \quad B(2)=3$$

Money in a bank account grows continuously at an annual rate of \(r\) (when the interest rate is \(5 \%, r=0.05,\) and so on). Suppose \(\$ 2000\) is put into the account in 2010 . (a) Write a differential equation satisfied by \(M,\) the amount of money in the account at time \(t,\) measured in years since 2010. (b) Solve the differential equation. (c) Sketch the solution until the year 2040 for interest rates of \(5 \%\) and \(10 \%\).

Morphine is administered to a patient intravenously at a rate of 2.5 mg per hour. About \(34.7 \%\) of the morphine is metabolized and leaves the body each hour. Write a differential equation for the amount of morphine, \(M,\) in milligrams, in the body as a function of time, \(t,\) in hours.

Check that \(y=A+C e^{k t}\) is a solution to the differential equation. $$\frac{d y}{d t}=k(y-A)$$

A bank account that earns \(10 \%\) interest compounded continuously has an initial balance of zero. Money is deposited into the account at a continuous rate of \(\$ 1000\) per year. (a) Write a differential equation that describes the rate of change of the balance \(B=f(t)\). (b) Solve the differential equation.

Alcohol is metabolized and excreted from the body at a rate of about one ounce of alcohol every hour. If some alcohol is consumed, write a differential equation for the amount of alcohol, \(A\) (in ounces), remaining in the body as a function of \(t,\) the number of hours since the alcohol was consumed.

A bank account earns \(5 \%\) annual interest, compounded continuously. Money is deposited in a continuous cash flow at a rate of \(\$ 1200\) per year into the account.\( (a) Write a differential equation that describes the rate at which the balance \)B=f(t)\( is changing. (b) Solve the differential equation given an initial balance \)B_{0}=0$. (c) Find the balance after 5 years.

(a) Consider the slope field for \(d y / d x=x y .\) What is the slope of the line segment at the point (2,1)\(?\) At (0,2)\(?\) At (-1,1)\(?\) At (2,-2)\(?\) (b) Sketch part of the slope field by drawing line segments with the slopes calculated in part (a).

The amount of ozone, \(Q,\) in the atmosphere is decreasing at a rate proportional to the amount of ozone present. If time \(t\) is measured in years, the constant of proportionality is \(-0.0025 .\) Write a differential equation for \(Q\) as a function of \(t,\) and give the general solution for the differential equation. If this rate continues, approximately what percent of the ozone in the atmosphere now will decay in the next 20 years?

Toxins in pesticides can get into the food chain and accumulate in the body. A person consumes 10 micrograms a day of a toxin, ingested throughout the day. The toxin leaves the body at a continuous rate of \(3 \%\) every day. Write a differential equation for the amount of toxin, \(A\) in micrograms, in the person's body as a function of the number of days, \(t\)

Money in an account earns interest at a continuous rate of 8\% per year, and payments are made continuously out of the account at the rate of $$\$ 5000\( a year. The account initially contains $$\$ 50,000 .\) Write a differential equation for the amount of money in the account, \(B\), in \(t\) years. Solve the differential equation. Does the account ever run out of money? If so, when?

A cup of coffee contains about 100 mg of caffeine. Caffeine is metabolized and leaves the body at a continuous rate of about \(17 \%\) every hour. (a) Write a differential equation for the amount, \(A,\) of caffeine in the body as a function of the number of hours, \(t,\) since the coffee was consumed. (b) Use the differential equation to find \(d A / d t\) at the start of the first hour (right after the coffee is consumed). Use your answer to estimate the change in the amount of caffeine during the first hour.

A company earns \(2 \%\) per month on its assets, paid continuously, and its expenses are paid out continuously at a rate of \(\$ 80,000\) per month.\( (a) Write a differential equation for the value, \)V\(, of the company as a function of time, \)t,\( in months. (b) What is the equilibrium solution for the differential equation? What is the significance of this value for the company? (c) Solve the differential equation found in part (a). (d) If the company has assets worth \)\$ 3\( million at time \)t=0,$ what are its assets worth one year later?

A person deposits money into an account at a continuous rate of $$ 6000\( a year, and the account earns interest at a continuous rate of \)7 \%\( per year. (a) Write a differential equation for the balance in the account, \)B\(, in dollars, as a function of years, \)t\( (b) Use the differential equation to calculate \)d B / d t\( if \)B=10,000\( and if \)B=100,000 .$ Interpret your answers.

A bank account earns \(7 \%\) annual interest compounded continuously. You deposit \(\$ 10,000 in the account, and withdraw money continuously from the account at a rate of \)\$ 1000\( per year.\) (a) Write a differential equation for the balance, \(B\), in the account after \(t\) years. (b) What is the equilibrium solution to the differential equation? (This is the amount that must be deposited now for the balance to stay the same over the years.) (c) Find the solution to the differential equation. (d) How much is in the account after 5 years? (e) Graph the solution. What happens to the balance in the long run?

A quantity \(W\) satisfies the differential equation $$\frac{d W}{d t}=5 W-20$$ (a) Is \(W\) increasing or decreasing at \(W=10 ? W=2 ?\) (b) For what values of \(W\) is the rate of change of \(W\) equal to zero?

In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For example, this is the case as \(\delta\) glucono-lactone changes into gluconic acid. (a) Write a differential equation satisfied by \(y,\) the quantity of \(\delta\) -glucono-lactone present at time \(t\) (b) If 100 grams of \(\delta\) -glucono-lactone is reduced to 54.9 grams in one hour, how many grams will remain after 10 hours?

Fill in the missing values in Table 9.3 given that \(d y / d t=\) \(0.5 t .\) Assume the rate of growth, given by \(d y / d t,\) is approximately constant over each unit time interval.$$\begin{array}{c|c|c|c|c|c} \hline t & 0 & 1 & 2 & 3 & 4 \\ \hline y & 8 & & & & \\ \hline \end{array}$$