Chapter 7

An unknown radioactive element decays into non-radioactive substances. In 320 days the radioactivity of a sample decreases by 58 percent. (a) What is the half-life of the element? half-life: (b) How long will it take for a sample of \(100 \mathrm{mg}\) to decay to \(88 \mathrm{mg} ?\) time needed:

Solve the separable differential equation for \(u\) $$ \frac{d u}{d t}=e^{2 u+8 t} $$ Use the following initial condition: \(u(0)=13\). \(=\)

Newton's Law of Cooling says that the rate at which an object, such as a cup of coffee, cools is proportional to the difference in the object's temperature and room temperature. If \(T(t)\) is the object's temperature and \(T_{r}\) is room temperature, this law is expressed at $$ \frac{d T}{d t}=-k\left(T-T_{r}\right) $$ where \(k\) is a constant of proportionality. In this problem, temperature is measured in degrees Fahrenheit and time in minutes. a. Two calculus students, Alice and Bob, enter a \(70^{\circ}\) classroom at the same time. Each has a cup of coffee that is \(100^{\circ} .\) The differential equation for Alice has a constant of proportionality \(k=0.5,\) while the constant of proportionality for Bob is \(k=0.1 .\) What is the initial rate of change for Alice's coffee? What is the initial rate of change for Bob's coffee? b. What feature of Alice's and Bob's cups of coffee could explain this difference? c. As the heating unit turns on and off in the room, the temperature in the room is $$ T_{r}=70+10 \sin t $$ Implement Euler's method with a step size of \(\Delta t=0.1\) to approximate the temperature of Alice's coffee over the time interval \(0 \leq t \leq 50 .\) This will most easily be performed using a spreadsheet such as Excel. Graph the temperature of her coffee and room temperature over this interval. d. In the same way, implement Euler's method to approximate the temperature of Bob's coffee over the same time interval. Graph the temperature of his coffee and room temperature over the interval. e. Explain the similarities and differences that you see in the behavior of Alice's and Bob's cups of coffee.

Given the differential equation \(x^{\prime}(t)=x^{4}-5 x^{3}-2 x^{2}+24 x+0 .\) List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equations are stable, semi- stable, or unstable. (It helps to sketch the graph. xFunctions will plot functions as well as phase planes.)

The logistic equation may be used to model how a rumor spreads through a group of people. Suppose that \(p(t)\) is the fraction of people that have heard the rumor on day \(t\). The equation $$ \frac{d p}{d t}=0.2 p(1-p) $$ describes how \(p\) changes. Suppose initially that one-tenth of the people have heard the rumor; that is, \(p(0)=0.1\) a. What happens to \(p(t)\) after a very long time? b. Determine a formula for the function \(p(t)\). c. At what time is \(p\) changing most rapidly? d. How long does it take before \(80 \%\) of the people have heard the rumor?

Find an equation of the curve that satisfies $$ \frac{d y}{d x}=45 y x^{4} $$ and whose \(y\) -intercept is \(5 .\) \(y(x)=\)

We have seen that the error in approximating the solution to an initial value problem is proportional to \(\Delta t .\) That is, if \(E_{\Delta t}\) is the Euler's method approximation to the solution to an initial value problem at \(\bar{t},\) then $$ y(\bar{t})-E_{\Delta t} \approx K \Delta t $$ for some constant of proportionality \(K .\) In this problem, we will see how to use this fact to improve our estimates, using an idea called accelerated convergence. a. We will create a new approximation by assuming the error is exactly proportional to \(\Delta t,\) according to the formula $$ y(\bar{t})-E_{\Delta t}=K \Delta t . $$ Using our earlier results from the initial value problem \(d y / d t=y\) and \(y(0)=1\) with \(\Delta t=0.2\) and \(\Delta t=0.1,\) we have $$ \begin{array}{l} y(1)-2.4883=0.2 \mathrm{~K} \\ y(1)-2.5937=0.1 \mathrm{~K} \end{array} $$ This is a system of two linear equations in the unknowns \(y(1)\) and \(K .\) Solve this system to find a new approximation for \(y(1)\). (You may remember that the exact value is \(y(1)=\) \(e=2.71828 \ldots . .)\) b. Use the other data, \(E_{0.05}=2.6533\) and \(E_{0.025}=2.6851\) to do similar work as in (a) to obtain another approximation. Which gives the better approximation? Why do you think this is? c. Let's now study the initial value problem $$ \frac{d y}{d t}=t-y, y(0)=0 $$ Approximate \(y(0.3)\) by applying Euler's method to find approximations \(E_{0.1}\) and \(E_{0.05}\). Now use the idea of accelerated convergence to obtain a better approximation. (For the sake of comparison, you want to note that the actual value is \(y(0.3)=0.0408 .\) )

Congratulations, you just won the lottery! In one option presented to you, you will be paid one million dollars a year for the next 25 years. You can deposit this money in an account that will earn \(5 \%\) each year. a. Set up a differential equation that describes the rate of change in the amount of money in the account. Two factors cause the amount to grow-first, you are depositing one millon dollars per year and second, you are earning \(5 \%\) interest. b. If there is no amount of money in the account when you open it, how much money will you have in the account after 25 years? c. The second option presented to you is to take a lump sum of 10 million dollars, which you will deposit into a similar account. How much money will you have in that account after 25 years? d. Do you prefer the first or second option? Explain your thinking. e. At what time does the amount of money in the account under the first option overtake the amount of money in the account under the second option?

The mass of a radioactive sample decays at a rate that is proportional to its mass. a. Express this fact as a differential equation for the mass \(M(t)\) using \(k\) for the constant of proportionality. b. If the initial mass is \(M_{0}\), find an expression for the mass \(M(t)\). c. The half-life of the sample is the amount of time required for half of the mass to decay. Knowing that the half-life of Carbon-14 is 5730 years, find the value of \(k\) for a sample of Carbon-14. d. How long does it take for a sample of Carbon-14 to be reduced to one- quarter its original mass? e. Carbon-14 naturally occurs in our environment; any living organism takes in Carbon14 when it eats and breathes. Upon dying, however, the organism no longer takes in Carbon-14. Suppose that you find remnants of a pre-historic firepit. By analyzing the charred wood in the pit, you determine that the amount of Carbon-14 is only \(30 \%\) of the amount in living trees. Estimate the age of the firepit.

In this problem, we test further what it means for a function to be a solution to a given differential equation. a. Consider the differential equation $$ \frac{d y}{d t}=y-t $$ Determine whether the following functions are solutions to the given differential equation. $$ \begin{array}{l} \text { - } y(t)=t+1+2 e^{t} \\ \text { - } y(t)=t+1 \\ \text { - } y(t)=t+2 \end{array} $$ b. When you weigh bananas in a scale at the grocery store, the height \(h\) of the bananas is described by the differential equation $$ \frac{d^{2} h}{d t^{2}}=-k h $$ where \(k\) is the spring constant, a constant that depends on the properties of the spring in the scale. After you put the bananas in the scale, you (cleverly) observe that the height of the bananas is given by \(h(t)=4 \sin (3 t) .\) What is the value of the spring constant?

Suppose that the population of a species of fish is controlled by the logistic equation $$ \frac{d P}{d t}=0.1 P(10-P) $$ where \(P\) is measured in thousands of fish and \(t\) is measured in years. a. What is the carrying capacity of this population? b. Suppose that a long time has passed and that the fish population is stable at the carrying capacity. At this time, humans begin harvesting \(20 \%\) of the fish every year. Modify the differential equation by adding a term to incorporate the harvesting of fish. c. What is the new carrying capacity? d. What will the fish population be one year after the harvesting begins? e. How long will it take for the population to be within \(10 \%\) of the carrying capacity?

When a skydiver jumps from a plane, gravity causes her downward velocity to increase at the rate of \(g \approx 9.8\) meters per second squared. At the same time, wind resistance causes her velocity to decrease at a rate proportional to the velocity. a. Using \(k\) to represent the constant of proportionality, write a differential equation that describes the rate of change of the skydiver's velocity. b. Find any equilibrium solutions and decide whether they are stable or unstable. Your result should depend on \(k\). c. Suppose that the initial velocity is zero. Find the velocity \(v(t)\). d. A typical terminal velocity for a skydiver falling face down is 54 meters per second. What is the value of \(k\) for this skydiver? e. How long does it take to reach \(50 \%\) of the terminal velocity?

Consider the initial value problem $$ \frac{d y}{d t}=-\frac{t}{y}, y(0)=8 $$ a. Find the solution of the initial value problem and sketch its graph. b. For what values of \(t\) is the solution defined? c. What is the value of \(y\) at the last time that the solution is defined? d. By looking at the differential equation, explain why we should not expect to find solutions with the value of \(y\) you noted in (c).

The population of a species of fish in a lake is \(P(t)\) where \(P\) is measured in thousands of fish and \(t\) is measured in months. The growth of the population is described by the differential equation $$ \frac{d P}{d t}=f(P)=P(6-P) $$ a. Sketch a graph of \(f(P)=P(6-P)\) and use it to determine the equilibrium solutions and whether they are stable or unstable. Write a complete sentence that describes the long-term behavior of the fish population. b. Suppose now that the owners of the lake allow fishers to remove 1000 fish from the lake every month (remember that \(P(t)\) is measured in thousands of fish). Modify the differential equation to take this into account. Sketch the new graph of \(d P / d t\) versus P. Determine the new equilibrium solutions and decide whether they are stable or unstable. c. Given the situation in part (b), give a description of the long-term behavior of the fish population. d. Suppose that fishermen remove \(h\) thousand fish per month. How is the differential equation modified? e. What is the largest number of fish that can be removed per month without eliminating the fish population? If fish are removed at this maximum rate, what is the eventual population of fish?

During the first few years of life, the rate at which a baby gains weight is proportional to the reciprocal of its weight. a. Express this fact as a differential equation. b. Suppose that a baby weighs 8 pounds at birth and 9 pounds one month later. How much will he weigh at one year? c. Do you think this is a realistic model for a long time?

Suppose that a cylindrical water tank with a hole in the bottom is filled with water. The water, of course, will leak out and the height of the water will decrease. Let \(h(t)\) denote the height of the water. A physical principle called Torricelli's Law implies that the height decreases at a rate proportional to the square root of the height. a. Express this fact using \(k\) as the constant of proportionality. b. Suppose you have two tanks, one with \(k=-1\) and another with \(k=-10 .\) What physical differences would you expect to find? c. Suppose you have a tank for which the height decreases at 20 inches per minute when the water is filled to a depth of 100 inches. Find the value of \(k\). d. Solve the initial value problem for the tank in part (c), and graph the solution you determine. e. How long does it take for the water to run out of the tank? f. Is the solution that you found valid for all time \(t ?\) If so, explain how you know this. If not, explain why not.

Let \(y(t)\) be the number of thousands of mice that live on a farm; assume time \(t\) is measured in years. \({ }^{1}\) a. The population of the mice grows at a yearly rate that is twenty times the number of mice. Express this as a differential equation. b. At some point, the farmer brings \(C\) cats to the farm. The number of mice that the cats can eat in a year is $$ M(y)=C \frac{y}{2+y} $$ thousand mice per year. Explain how this modifies the differential equation that you found in part a). c. Sketch a graph of the function \(M(y)\) for a single cat \(C=1\) and explain its features by looking, for instance, at the behavior of \(M(y)\) when \(y\) is small and when \(y\) is large. d. Suppose that \(C=1\). Find the equilibrium solutions and determine whether they are stable or unstable. Use this to explain the long-term behavior of the mice population depending on the initial population of the mice. e. Suppose that \(C=60 .\) Find the equilibrium solutions and determine whether they are stable or unstable. Use this to explain the long-term behavior of the mice population depending on the initial population of the mice. f. What is the smallest number of cats you would need to keep the mice population from growing arbitrarily large?

Suppose that you have a water tank that holds 100 gallons of water. A briny solution, which contains 20 grams of salt per gallon, enters the tank at the rate of 3 gallons per minute. At the same time, the solution is well mixed, and water is pumped out of the tank at the rate of 3 gallons per minute. a. Since 3 gallons enters the tank every minute and 3 gallons leaves every minute, what can you conclude about the volume of water in the tank. b. How many grams of salt enters the tank every minute? c. Suppose that \(S(t)\) denotes the number of grams of salt in the tank in minute \(t\). How many grams are there in each gallon in minute \(t\) ? d. Since water leaves the tank at 3 gallons per minute, how many grams of salt leave the tank each minute? e. Write a differential equation that expresses the total rate of change of \(S\). f. Identify any equilibrium solutions and determine whether they are stable or unstable. g. Suppose that there is initially no salt in the tank. Find the amount of salt \(S(t)\) in minute \(t .\) h. What happens to \(S(t)\) after a very long time? Explain how you could have predicted this only knowing how much salt there is in each gallon of the briny solution that enters the tank.

The Gompertz equation is a model that is used to describe the growth of certain populations. Suppose that \(P(t)\) is the population of some organism and that $$ \frac{d P}{d t}=-P \ln \left(\frac{P}{3}\right)=-P(\ln P-\ln 3) $$ a. Sketch a slope field for \(P(t)\) over the range \(0 \leq P \leq 6\). b. Identify any equilibrium solutions and determine whether they are stable or unstable. c. Find the population \(P(t)\) assuming that \(P(0)=1\) and sketch its graph. What happens to \(P(t)\) after a very long time? d. Find the population \(P(t)\) assuming that \(P(0)=6\) and sketch its graph. What happens to \(P(t)\) after a very long time? e. Verify that the long-term behavior of your solutions agrees with what you predicted by looking at the slope field.