Answer: We have not discussed methods by which systems of first-order differential

We have not discussed methods by which systems of first-order differential equations can be solved. Nevertheless, systems such as (2) can be solved with no knowledge other than how to solve a single linear firstorder equation. Find a solution of (2) subject to the initial conditions x(0)  x0, y(0)  0, z(0)  0.

In Problem 1 suppose that time is measured in days, that the decay constants are k1 
0.138629 and k2 
0.004951, and that x0  20. Use a graphing utility to obtain the graphs of the solutions x(t), y(t), and z(t) on the same set of coordinate axes. Use the graphs to approximate the half-lives of substances X and Y.

Use the graphs in Problem 2 to approximate the times when the amounts x(t) and y(t) are the same, the times when the amounts x(t) and z(t) are the same, and the times when the amounts y(t) and z(t) are the same. Why does the time that is determined when the amounts y(t) and z(t) are the same make intuitive sense?

Construct a mathematical model for a radioactive series of four elements W, X, Y, and Z, where Z is a stable element. Mixtures

Consider two tanks A and B, with liquid being pumped in and out at the same rates, as described by the system of equations (3). What is the system of differential equations if, instead of pure water, a brine solution containing 2 pounds of salt per gallon is pumped into tank A?

Use the information given in Figure 3.3.5 to construct a mathematical model for the number of pounds of salt x1(t), x2(t), and x3(t) at time t in tanks A, B, and C, respectively.

Two very large tanks A and B are each partially filled with 100 gallons of brine. Initially, 100 pounds of salt is dissolved in the solution in tank A and 50 pounds of salt is dissolved in the solution in tank B. The system is closed in that the well-stirred liquid is pumped only between the tanks, as shown in Figure 3.3.6. (b) Find a relationship between the variables x1(t) and x2(t) that holds at time t. Explain why this relationship makes intuitive sense. Use this relationship to help find the amount of salt in tank B at t  30 min.

Three large tanks contain brine, as shown in Figure 3.3.7. Use the information in the figure to construct a mathematical model for the number of pounds of salt x1(t), x2(t), and x3(t) at time t in tanks A, B, and C, respectively. Without solving the system, predict limiting values of x1(t), x2(t), and x3(t) as t : .

Consider the Lotka-Volterra predator-prey model defined by , where the populations x(t) (predators) and y(t) (prey) are measured in thousands. Suppose x(0)  6 and y(0)  6. Use a numerical solver to graph x(t) and y(t). Use the graphs to approximate the time t  0 when the two populations are first equal. Use the graphs to approximate the period of each population.

Consider the competition model defined by , where the populations x(t) and y(t) are measured in thousands and t in years. Use a numerical solver to analyze the populations over a long period of time for each of the following cases: (a) x(0)  1.5, y(0)  3.5 (b) x(0)  1, y(0)  1 (c) x(0)  2, y(0)  7 (d) x(0)  4.5, y(0)  0.5

Consider the competition model defined by , where the populations x(t) and y(t) are measured in thousands and t in years. Use a numerical solver to analyze the populations over a long period of time for each of the following cases: (a) x(0)  1, y(0)  1 (b) x(0)  4, y(0)  10 (c) x(0)  9, y(0)  4 (d) x(0)  5.5, y(0)  3.5

Show that a system of differential equations that describes the currents i2(t) and i3(t) in the electrical network shown in Figure 3.3.8 is

Determine a system of first-order differential equations that describes the currents i2(t) and i3(t) in the electrical network shown in Figure 3.3.9.

Show that the linear system given in (18) describes the currents i1(t) and i2(t) in the network shown in Figure 3.3.4. [

SIR Model A communicable disease is spread throughout a small community, with a fixed population of n people, by contact between infected individuals and people who are susceptible to the disease. Suppose that everyone is initially susceptible to the disease and that no one leaves the community while the epidemic is spreading. At time t, let s(t), i(t), and r(t) denote, in turn, the number of people in the community (measured in hundreds) who are susceptible to the disease but not yet infected with it, the number of people who are infected with the disease, and the number of people who have recovered from the disease. Explain why the system of differential equations where k1 (called the infection rate) and k2 (called the removal rate) are positive constants, is a reasonable mathematical model, commonly called a SIR model, for the spread of the epidemic throughout the community. Give plausible initial conditions associated with this system of equations.

(a) In Problem 15, explain why it is sufficient to analyze only . (b) Suppose k1  0.2, k2  0.7, and n  10. Choose various values of i(0)  i0, 0  i0  10. Use a numerical solver to determine what the model predicts about the epidemic in the two cases s0  k2
k1 and s0  k2
k1. In the case of an epidemic, estimate the number of people who are eventually infected.

Concentration of a Nutrient Suppose compartments A and B shown in Figure 3.3.10 are filled with fluids andare separated by a permeable membrane. The figure is a compartmental representation of the exterior and interior of a cell. Suppose, too, that a nutrient necessary for cell growth passes through the membrane. A model for the concentrations x(t) and y(t) of the nutrient in compartments A and B, respectively, at time t is given by the linear system of differential equations , where VA and VB are the volumes of the compartments, and k  0 is a permeability factor. Let x(0)  x0 and y(0)  y0 denote the initial concentrations of the nutrient. Solely on the basis of the equations in the system and the assumption x0  y0  0, sketch, on the same set of coordinate axes, possible solution curves of the system. Explain your reasoning. Discuss the behavior of the solutions over a long period of time.

The system in Problem 17, like the system in (2), can be solved with no advanced knowledge. Solve for x(t) and y(t) and compare their graphs with your sketches in Problem 17. Determine the limiting values of x(t) and y(t) as t : . Explain why the answer to the last question makes intuitive sense.

Solely on the basis of the physical description of the mixture problem on page 107 and in Figure 3.3.1, discuss the nature of the functions x1(t) and x2(t). What is the behavior of each function over a long period of time? Sketch possible graphs of x1(t) and x2(t). Check your conjectures by using a numerical solver to obtain numerical solution curves of (3) subject to the initial conditions x1(0)  25, x2(0)  0.

Newtons Law of Cooling/Warming As shown in Figure 3.3.11, a small metal bar is placed inside container A, and container A then is placed within a much larger container B. As the metal bar cools, the ambient temperature TA(t) of the medium within container A changes according to Newtons law of cooling. As container A cools, the temperature of the medium inside container B does not change significantly and can be considered to be a constant TB. Construct a mathematical model for the temperatures T(t) and TA(t), where T(t) is the temperature of the metal bar inside container A. As in Problems 1 and 18, this model can be solved by using prior knowledge. Find a solution of the system subject to the initial conditions T(0)  T0, TA(0)  T1.

A Mixing Problem A pair of tanks are connected as shown in Figure 3.3.12. At t  0, tank A contains 500 liters of liquid, 200 of which are ethanol, and tank B contains 100 liters of liquid, 7 of which are ethanol. Beginning at t  0, 3 liters of 20% ethanol solution are added per minute. An additional 2 L/min are pumped from tank B back into tank A. The result is continuously mixed, and 5 L/min are pumped into tank B. The contents of tank B are also continuously mixed. In addition to the 2 liters that are returned to tank A, 3 L/min are discharged from the system. Let P(t) and Q(t) denote the number of liters of ethanol in tanks A and B at time t. We wish to find P(t). Using the principle that rate of change  input rate of ethanol
output rate of ethanol, we obtain the system of first-order differential equations (19) (20) (a) Qualitatively discuss the behavior of the system. What is happening in the short term? What happens in the long term? dQ dt  5 P 500
5 Q 100  P 100
Q 20. dP dt  3(0.2) 2 Q 100
5 P 500  0.6 Q 50
P 100 CHAPTER 3 IN REVIEW 113 Michael Prophet, Ph.D Doug Shaw, Ph.D Associate Professors Mathematics Department University of Northern Iowa FIGURE 3.3.12 Mixing tanks in Problem 21 mixture 3 L/min mixture 2 L/min B 100 liters A 500 liters ethanol solution 3 L/min mixture 5 L/min (b) We now attempt to solve this system. When (19) is differentiated with respect to t, we obtain Substitute (20) into this equation and simplify. (c) Show that when we solve (19) for Q and substitute it into our answer in part (b), we obtain (d) We are given that P(0)  200. Show that Then solve the differential equation in part (c) subject to these initial conditions. (e) Substitute the solution of part (d) back into (19) and solve for Q(t). (f) What happens to P(t) and Q(t) as ? t : P