Determine a system of first-order differential equations

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 first order 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.

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. mixture 2 gal/min mixture 3 gal/min B 100 gal A 100 gal FIGURE 3.3.6 Mixing tanks in Problem 7 (a) Use the information given in the figure to construct a mathematical model for the number of pounds of salt x1(t) and x2(t) at time t in tanks A and B, respectively. (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), FIGURE 3.3.7 Mixing tanks in Problem 8

Consider the Lotka-Volterra predator-prey model defined b , 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. C

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 11. Co

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 dy dt y(1.7  0.1y  0.15x) dx dt x(1  0.1x  0.05y) dy dt y(1  0.1y  0.3x) dx dt x(2  0.4x  0.3y) dy dt 0.2y  0.025xy dx dt 0.1x  0.02xy 27069_03_ch03_p083-115.qxd 2/2/12 2:32 PM Page 111analyze 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 Networ

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 R1 di2 dt  R2 di3 dt  1 C i3 0. L di2 dt  L di3 dt  R1i2 E(t)

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. [Hint: dqdt i3.]

SIR Model Acommunicable disease is spread throughout a small community, with a fixe 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 k2k1 and s0 k2k1. In the case of an epidemic, estimate the number of people who are eventually infected. Proje

Concentration of a Nutrient Suppose compartments A and B shown in Figure 3.3.10 are filled with fluids and are 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

Mixtures 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. FIGURE 3.3.11 Container within a container in Problem 20