INTRODUCTION

In this section we introduce the notion of a mathematical model. Roughly speaking, a mathematical model is a mathematical description of something. This description could be as simple as a function. For example, Leonardo da Vinci (1452–1519) was able to deduce the speed v of a falling body by examining a sequence. Leonardo allowed water drops to fall, at equally spaced intervals of time, between two boards covered with blotting paper. When a spring mechanism was disengaged, the boards were clapped together. See FIGURE 1.3.1. By carefully examining the sequence of water blots, Leonardo discovered that the distances between consecutive drops increased in “a continuous arithmetic proportion.” In this manner he discovered the formula v = gt.

Although there are many kinds of mathematical models, in this section we focus only on differential equations and discuss some specific differential-equation models in biology, physics, and chemistry. Once we have studied some methods for solving DEs, in Chapters 2 and 3 we return to, and solve, some of these models.

Mathematical Models

It is often desirable to describe the behavior of some real-life system or phenomenon, whether physical, sociological, or even economic, in mathematical terms. The mathematical description of a system or a phenomenon is called a mathematical model and is constructed with certain goals in mind. For example, we may wish to understand the mechanisms of a certain ecosystem by studying the growth of animal populations in that system, or we may wish to date fossils by means of analyzing the decay of a radioactive substance either in the fossil or in the stratum in which it was discovered.

Construction of a mathematical model of a system starts with identification of the variables that are responsible for changing the system. We may choose not to incorporate all these variables into the model at first. In this first step we are specifying the level of resolution of the model. Next, we make a set of reasonable assumptions or hypotheses about the system we are trying to describe. These assumptions will also include any empirical laws that may be applicable to the system.

For some purposes it may be perfectly within reason to be content with low-resolution models. For example, you may already be aware that in modeling the motion of a body falling near the surface of the Earth, the retarding force of air friction is sometimes ignored in beginning physics courses; but if you are a scientist whose job it is to accurately predict the flight path of a long-range projectile, air resistance and other factors such as the curvature of the Earth have to be taken into account.

Since the assumptions made about a system frequently involve a rate of change of one or more of the variables, the mathematical depiction of all these assumptions may be one or more equations involving derivatives. In other words, the mathematical model may be a differential equation or a system of differential equations.

Once we have formulated a mathematical model that is either a differential equation or a system of differential equations, we are faced with the not insignificant problem of trying to solve it. If we can solve it, then we deem the model to be reasonable if its solution is consistent with either experimental data or known facts about the behavior of the system. But if the predictions produced by the solution are poor, we can either increase the level of resolution of the model or make alternative assumptions about the mechanisms for change in the system. The steps of the modeling process are then repeated as shown in FIGURE 1.3.2.

Of course, by increasing the resolution we add to the complexity of the mathematical model and increase the likelihood that we cannot obtain an explicit solution.

A mathematical model of a physical system will often involve the variable time t. A solution of the model then gives the state of the system; in other words, for appropriate values of t, the values of the dependent variable (or variables) describe the system in the past, present, and future.

Population Dynamics

One of the earliest attempts to model human population growth by means of mathematics was by the English economist Thomas Robert Malthus (1776–1834) in 1798. Basically, the idea of the Malthusian model is the assumption that the rate at which a population of a country grows at a certain time is proportional^* to the total population of the country at that time. In other words, the more people there are at time t, the more there are going to be in the future. In mathematical terms, if P(t) denotes the total population at time t, then this assumption can be expressed as

or (1)

where k is a constant of proportionality. This simple model, which fails to take into account many factors (immigration and emigration, for example) that can influence human populations to either grow or decline, nevertheless turned out to be fairly accurate in predicting the population of the United States during the years 1790–1860. Populations that grow at a rate described by (1) are rare; nevertheless, (1) is still used to model growth of small populations over short intervals of time, for example, bacteria growing in a petri dish.

Radioactive Decay

The nucleus of an atom consists of combinations of protons and neutrons. Many of these combinations of protons and neutrons are unstable; that is, the atoms decay or transmute into the atoms of another substance. Such nuclei are said to be radioactive. For example, over time, the highly radioactive radium, Ra-226, transmutes into the radioactive gas radon, Rn-222. In modeling the phenomenon of radioactive decay, it is assumed that the rate dA/dt at which the nuclei of a substance decay is proportional to the amount (more precisely, the number of nuclei) A(t) of the substance remaining at time t:

or (2)

Of course equations (1) and (2) are exactly the same; the difference is only in the interpretation of the symbols and the constants of proportionality. For growth, as we expect in (1), k > 0, and in the case of (2) and decay, k < 0.

The model (1) for growth can be seen as the equation dS/dt = rS, which describes the growth of capital S when an annual rate of interest r is compounded continuously. The model (2) for decay also occurs in a biological setting, such as determining the half-life of a drug—the time that it takes for 50% of a drug to be eliminated from a body by excretion or metabolism. In chemistry, the decay model (2) appears as the mathematical description of a first-order chemical reaction. The point is this:

Mathematical models are often accompanied by certain side conditions. For example, in (1) and (2) we would expect to know, in turn, an initial population P₀ and an initial amount of radioactive substance A₀ that is on hand. If this initial point in time is taken to be t = 0, then we know that P(0) = P₀ and A(0) = A₀. In other words, a mathematical model can consist of either an initial-value problem or, as we shall see later in Section 3.9, a boundary-value problem.

Newton’s Law of Cooling/Warming

According to Newton’s empirical law of cooling—or warming—the rate at which the temperature of a body changes is proportional to the difference between the temperature of the body and the temperature of the surrounding medium, the so-called ambient temperature. If T(t) represents the temperature of a body at time t, T_m the temperature of the surrounding medium, and dT/dt the rate at which the temperature of the body changes, then Newton’s law of cooling/warming translates into the mathematical statement

or (3)

where k is a constant of proportionality. In either case, cooling or warming, if T_m is a constant, it stands to reason that k < 0.

Spread of a Disease

A contagious disease—for example, a flu virus—is spread throughout a community by people coming into contact with other people. Let x(t) denote the number of people who have contracted the disease and y(t) the number of people who have not yet been exposed. It seems reasonable to assume that the rate dx/dt at which the disease spreads is proportional to the number of encounters or interactions between these two groups of people. If we assume that the number of interactions is jointly proportional to x(t) and y(t), that is, proportional to the product xy, then

(4)

where k is the usual constant of proportionality. Suppose a small community has a fixed population of n people. If one infected person is introduced into this community, then it could be argued that x(t) and y(t) are related by x + y = n + 1. Using this last equation to eliminate y in (4) gives us the model

.(5)

An obvious initial condition accompanying equation (5) is x(0) = 1.

Chemical Reactions

The disintegration of a radioactive substance, governed by the differential equation (2), is said to be a first-order reaction. In chemistry, a few reactions follow this same empirical law: If the molecules of substance A decompose into smaller molecules, it is a natural assumption that the rate at which this decomposition takes place is proportional to the amount of the first substance that has not undergone conversion; that is, if X(t) is the amount of substance A remaining at any time, then dX/dt = kX, where k is a negative constant since X is decreasing. An example of a first-order chemical reaction is the conversion of t-butyl chloride into t-butyl alcohol:

(CH₃)₃CCl + NaOH → (CH₃)₃COH + NaCl.

Only the concentration of the t-butyl chloride controls the rate of reaction. But in the reaction

CH₃Cl + NaOH → CH₃OH + NaCl,

for every molecule of methyl chloride, one molecule of sodium hydroxide is consumed, thus forming one molecule of methyl alcohol and one molecule of sodium chloride. In this case the rate at which the reaction proceeds is proportional to the product of the remaining concentrations of CH₃Cl and of NaOH. If X denotes the amount of CH₃OH formed and α and β are the given amounts of the first two chemicals A and B, then the instantaneous amounts not converted to chemical C are α − X and β − X, respectively. Hence the rate of formation of C is given by

,(6)

where k is a constant of proportionality. A reaction whose model is equation (6) is said to be second order.

Mixtures

The mixing of two salt solutions of differing concentrations gives rise to a first-order differential equation for the amount of salt contained in the mixture. Let us suppose that a large mixing tank initially holds 300 gallons of brine (that is, water in which a certain number of pounds of salt has been dissolved). Another brine solution is pumped into the large tank at a rate of 3 gallons per minute; the concentration of the salt in this inflow is 2 pounds of salt per gallon. When the solution in the tank is well stirred, it is pumped out at the same rate as the entering solution. See FIGURE 1.3.3. If A(t) denotes the amount of salt (measured in pounds) in the tank at time t, then the rate at which A(t) changes is a net rate:

(7)

The input rate R_in at which the salt enters the tank is the product of the inflow concentration of salt and the inflow rate of fluid. Note that R_in is measured in pounds per minute:

Now, since the solution is being pumped out of the tank at the same rate that it is pumped in, the number of gallons of brine in the tank at time t is a constant 300 gallons. Hence the concentration of the salt in the tank, as well as in the outflow, is c(t) = A(t)/300 lb/gal, and so the output rate R_out of salt is

The net rate (7) then becomes

or .(8)

If r_in and r_out denote general input and output rates of the brine solutions,^* respectively, then there are three possibilities: r_in = r_out, r_in > r_out, and r_in < r_out. In the analysis leading to (8) we have assumed that r_in = r_out. In the latter two cases, the number of gallons of brine in the tank is either increasing (r_in > r_out) or decreasing (r_in < r_out) at the net rate r_in − r_out. See Problems 10–12 in Exercises 1.3.

Draining a Tank

Evangelista Torricelli (1608–1647) was an Italian physicist who invented the barometer and was a student of Galileo Galilei. In hydrodynamics, Torricelli’s law states that the speed v of efflux of water through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h; that is, , where g is the acceleration due to gravity. This last expression comes from equating the kinetic energy mv² with the potential energy mgh and solving for v. Suppose a tank filled with water is allowed to drain through a hole under the influence of gravity. We would like to find the depth h of water remaining in the tank at time t. Consider the tank shown in FIGURE 1.3.4. If the area of the hole is A_h (in ft²) and the speed of the water leaving the tank is (in ft/s), then the volume of water leaving the tank per second is (in ft³/s). Thus if V(t) denotes the volume of water in the tank at time t,

,(9)

where the minus sign indicates that V is decreasing. Note here that we are ignoring the possibility of friction at the hole that might cause a reduction of the rate of flow there. Now if the tank is such that the volume of water in it at time t can be written V(t) = A_wh, where A_w (in ft²) is the constant area of the upper surface of the water (see Figure 1.3.4), then dV/dt = A_wdh/dt. Substituting this last expression into (9) gives us the desired differential equation for the height of the water at time t:

.(10)

It is interesting to note that (10) remains valid even when A_w is not constant. In this case we must express the upper surface area of the water as a function of h; that is, A_w = A(h). See Problem 14 in Exercises 1.3.

Series Circuits

The mathematical analysis of electrical circuits and networks is relatively straightforward, using two laws formulated by the German physicist Gustav Robert Kirchhoff (1824–1887) in 1845 while he was still a student. Consider the single-loop LRC-series circuit containing an inductor, resistor, and capacitor shown in FIGURE 1.3.5(a). The current in a circuit after a switch is closed is denoted by i(t); the charge on a capacitor at time t is denoted by q(t). The letters L, R, and C are known as inductance, resistance, and capacitance, respectively, and are generally constants. Now according to Kirchhoff’s second law, the impressed voltage E(t) on a closed loop must equal the sum of the voltage drops in the loop. Figure 1.3.5(b) also shows the symbols and the formulas for the respective voltage drops across an inductor, a resistor, and a capacitor. Since current i(t) is related to charge q(t) on the capacitor by i = dq/dt, by adding the three voltage drops

,

and equating the sum to the impressed voltage, we obtain a second-order differential equation

.(11)

We will examine a differential equation analogous to (11) in great detail in Section 3.8.

Falling Bodies

In constructing a mathematical model of the motion of a body moving in a force field, one often starts with Newton’s second law of motion. Recall from elementary physics that Newton’s first law of motion states that a body will either remain at rest or will continue to move with a constant velocity unless acted upon by an external force. In each case this is equivalent to saying that when the sum of the forces F = ∑F_k—that is, the net or resultant force—acting on the body is zero, then the acceleration a of the body is zero. Newton’s second law of motion indicates that when the net force acting on a body is not zero, then the net force is proportional to its acceleration a, or more precisely, F = ma, where m is the mass of the body.

Now suppose a rock is tossed upward from a roof of a building as illustrated in FIGURE 1.3.6. What is the position s(t) of the rock relative to the ground at time t? The acceleration of the rock is the second derivative d²s/dt². If we assume that the upward direction is positive and that no force acts on the rock other than the force of gravity, then Newton’s second law gives

or .(12)

In other words, the net force is simply the weight F = F₁ = −W of the rock near the surface of the Earth. Recall that the magnitude of the weight is W = mg, where m is the mass of the body and g is the acceleration due to gravity. The minus sign in (12) is used because the weight of the rock is a force directed downward, which is opposite to the positive direction. If the height of the building is s₀ and the initial velocity of the rock is v₀, then s is determined from the second-order initial-value problem

, , .(13)

Although we have not stressed solutions of the equations we have constructed, we note that (13) can be solved by integrating the constant −g twice with respect to t. The initial conditions determine the two constants of integration. You might recognize the solution of (13) from elementary physics as the formula s(t) = gt² + v₀t + s₀.

Falling Bodies and Air Resistance

Prior to the famous experiment by Italian mathematician and physicist Galileo Galilei (1564–1642) from the Leaning Tower of Pisa, it was generally believed that heavier objects in free fall, such as a cannonball, fell with a greater acceleration than lighter objects, such as a feather. Obviously a cannonball and a feather, when dropped simultaneously from the same height, do fall at different rates, but it is not because a cannonball is heavier. The difference in rates is due to air resistance. The resistive force of air was ignored in the model given in (13). Under some circumstances a falling body of mass m—such as a feather with low density and irregular shape—encounters air resistance proportional to its instantaneous velocity v. If we take, in this circumstance, the positive direction to be oriented downward, then the net force acting on the mass is given by F = F₁ + F₂ = mg − kv, where the weight F₁ = mg of the body is a force acting in the positive direction and air resistance F₂ = −kv is a force, called viscous damping, or drag, acting in the opposite or upward direction. See FIGURE 1.3.7. Now since v is related to acceleration a by a = dv/dt, Newton’s second law becomes F = ma = mdv/dt. By equating the net force to this form of Newton’s second law, we obtain a first-order differential equation for the velocity v(t) of the body at time t,

.(14)

Here k is a positive constant of proportionality called the drag coefficient. If s(t) is the distance the body falls in time t from its initial point of release, then v = ds/dt and a = dv/dt = d²s/dt². In terms of s, (14) is a second-order differential equation

or .(15)

Suspended Cables

Suppose a flexible cable, wire, or heavy rope is suspended between two vertical supports. Physical examples of this could be a long telephone wire strung between two posts as shown in red in FIGURE 1.3.8(a), or one of the two cables supporting the roadbed of a suspension bridge shown in red in Figure 1.3.8(b). Our goal is to construct a mathematical model that describes the shape that such a cable assumes.

To begin, let’s agree to examine only a portion or element of the cable between its lowest point P₁ and any arbitrary point P₂. As drawn in blue in FIGURE 1.3.9, this element of the cable is the curve in a rectangular coordinate system with the y-axis chosen to pass through the lowest point P₁ on the curve and the x-axis chosen a units below P₁. Three forces are acting on the cable: the tensions T₁ and T₂ in the cable that are tangent to the cable at P₁ and P₂, respectively, and the portion W of the total vertical load between the points P₁ and P₂. Let T₁ = |T₁|, T₂ = |T₂|, and W = |W| denote the magnitudes of these vectors. Now the tension T₂ resolves into horizontal and vertical components T₂ cos θ and T₂ sin θ. Because of static equilibrium, we can write

and .

By dividing the last equation by the first, we eliminate T₂ and get tan θ = W/T₁. But since dy/dx = tan θ, we arrive at

.(16)

This simple first-order differential equation serves as a model for both the shape of a flexible wire, such as a telephone wire hanging under its own weight, as well as the shape of the cables that support the roadbed. We will come back to equation (16) in Exercises 2.2 and in Section 3.11.

1.3 Exercises Answers to selected odd-numbered problems begin on page ANS-1.

Population Dynamics

1. Under the same assumptions underlying the model in (1), determine a differential equation governing the growing population P(t) of a country when individuals are allowed to immigrate into the country at a constant rate r > 0. What is the differential equation for the population P(t) of the country when individuals are allowed to emigrate at a constant rate r > 0?
2. The population model given in (1) fails to take death into consideration; the growth rate equals the birth rate. In another model of a changing population of a community, it is assumed that the rate at which the population changes is a net rate—that is, the difference between the rate of births and the rate of deaths in the community. Determine a model for the population P(t) if both the birth rate and the death rate are proportional to the population present at time t.
3. Using the concept of a net rate introduced in Problem 2, determine a differential equation governing a population P(t) if the birth rate is proportional to the population present at time t but the death rate is proportional to the square of the population present at time t.
4. Modify the model in Problem 3 for the net rate at which the population P(t) of a certain kind of fish changes by also assuming that the fish are harvested at a constant rate h > 0.

Newton’s Law of Cooling/Warming

5. A cup of coffee cools according to Newton’s law of cooling (3). Use data from the graph of the temperature T(t) in FIGURE 1.3.10 to estimate the constants T_m, T₀, and k in a model of the form of the first-order initial-value problem

,

6. The ambient temperature T_m in (3) could be a function of time t. Suppose that in an artificially controlled environment, T_m(t) is periodic with a 24-hour period, as illustrated in FIGURE 1.3.11. Devise a mathematical model for the temperature T(t) of a body within this environment.

Spread of a Disease/Technology

7. Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. Determine a differential equation governing the number of students x(t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between the number of students with the flu and the number of students who have not yet been exposed to it.
8. At a time t = 0, a technological innovation is introduced into a community with a fixed population of n people. Determine a differential equation governing the number of people x(t) who have adopted the innovation at time t if it is assumed that the rate at which the innovation spreads through the community is jointly proportional to the number of people who have adopted it and the number of people who have not adopted it.

Mixtures

9. Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt has been dissolved. Pure water is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is pumped out at the same rate. Determine a differential equation for the amount A(t) of salt in the tank at time t. What is A(0)?
10. Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt has been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is pumped out at a slower rate of 2 gal/min. If the concentration of the solution entering is 2 lb/gal, determine a differential equation for the amount A(t) of salt in the tank at time t.
11. What is the differential equation in Problem 10 if the well-stirred solution is pumped out at a faster rate of 3.5 gal/min?
12. Generalize the model given in (8) of this section by assuming that the large tank initially contains N₀ number of gallons of brine, r_in and r_out are the input and output rates of the brine, respectively (measured in gallons per minute), c_in is the concentration of the salt in the inflow, c(t) is the concentration of the salt in the tank as well as in the outflow at time t (measured in pounds of salt per gallon), and A(t) is the amount of salt in the tank at time t.

Draining a Tank

13. Suppose water is leaking from a tank through a circular hole of area A_h at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of the water leaving the tank per second to where c(0 < c < 1) is an empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank in FIGURE 1.3.12. The radius of the hole is 2 in. and g = 32 ft/s².
    

    

14. The right-circular conical tank shown in FIGURE 1.3.13 loses water out of a circular hole at its bottom. Determine a differential equation for the height of the water h at time t. The radius of the hole is 2 in., g = 32 ft/s², and the friction/contraction factor introduced in Problem 13 is c = 0.6.
    

    

Series Circuits

15. A series circuit contains a resistor and an inductor as shown in FIGURE 1.3.14. Determine a differential equation for the current i(t) if the resistance is R, the inductance is L, and the impressed voltage is E(t).
    

    

16. A series circuit contains a resistor and a capacitor as shown in FIGURE 1.3.15. Determine a differential equation for the charge q(t) on the capacitor if the resistance is R, the capacitance is C, and the impressed voltage is E(t).
    

    

Falling Bodies and Air Resistance

17. For high-speed motion through the air—such as the skydiver shown in FIGURE 1.3.16 falling before the parachute is opened—air resistance is closer to a power of the instantaneous velocity v(t). Determine a differential equation for the velocity v(t) of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity.
    

    

Newton’s Second Law and Archimedes’ Principle

18. A cylindrical barrel s ft in diameter of weight w lb is floating in water as shown in FIGURE 1.3.17(a). After an initial depression, the barrel exhibits an up-and-down bobbing motion along a vertical line. Using Figure 1.3.17(b), determine a differential equation for the vertical displacement y(t) if the origin is taken to be on the vertical axis at the surface of the water when the barrel is at rest. Assume the downward direction is positive, that the weight density of the water is 62.4 lb/ft³, and that there is no resistance between the barrel and the water. Use Archimedes’ principle: Buoyancy, or the upward force of the water on the barrel, is equal to the weight of the water displaced. Archimedes of Syracuse (287 BCE–212 BCE) was arguably one of the greatest scientists/mathematicians of antiquity. Using his approximation of the number π, he found the area of a circle as well as the surface area and volume of a sphere.
    

    

Newton’s Second Law and Hooke’s Law

19. After a mass m is attached to a spring, it stretches s units and then hangs at rest in the equilibrium position as shown in FIGURE 1.3.18(b). After the spring/mass system has been set in motion, let x(t) denote the directed distance of the mass beyond the equilibrium position. As indicated in Figure 1.3.18(c), assume that the downward direction is positive, that the motion takes place in a vertical straight line through the center of gravity of the mass, and that the only forces acting on the system are the weight of the mass and the restoring force of the stretched spring. Use Hooke’s law: The restoring force of a spring is proportional to its total elongation. Determine a differential equation for the displacement x(t) at time t.
    

    

20. In Problem 19, what is a differential equation for the displacement x(t) if the motion takes place in a medium that imparts a damping force on the spring/mass system that is proportional to the instantaneous velocity of the mass and acts in a direction opposite to that of motion?

Newton’s Second Law and Variable Mass

When the mass m of a body moving through a force field is variable, Newton’s second law of motion takes on the following form: If the net force acting on a body is not zero, then the net force F is equal to the time rate of change of momentum of the body. That is,

^*,(17)

where mv is momentum. Use this formulation of Newton’s second law in Problems 21 and 22.

21. Consider a single-stage rocket that is launched vertically upward as shown in the accompanying photo. Let m(t) denote the total mass of the rocket at time t (which is the sum of three masses: the constant mass of the payload, the constant mass of the vehicle, and the variable amount of fuel). Assume that the positive direction is upward, air resistance is proportional to the instantaneous velocity v of the rocket, and R is the upward thrust or force generated by the propulsion system. Use (17) to find a mathematical model for the velocity v(t) of the rocket.
    

    

22. In Problem 21, suppose m(t) = m_p + m_v + m_f(t) where m_p is constant mass of the payload, m_v is the constant mass of the vehicle, and m_f(t) is the variable amount of fuel.
    1. Show that the rate at which the total mass of the rocket changes is the same as the rate at which the mass of the fuel changes.
    2. If the rocket consumes its fuel at a constant rate λ, find m(t). Then rewrite the differential equation in Problem 21 in terms of λ and the initial total mass m(0) = m₀.
    3. Under the assumption in part (b), show that the burnout time t_b > 0 of the rocket, or the time at which all the fuel is consumed, is t_b = m_f(0)/λ, where m_f(0) is the initial mass of the fuel.

Newton’s Second Law and the Law of Universal Gravitation

23. By Newton’s law of universal gravitation, the free-fall acceleration a of a body, such as the satellite shown in FIGURE 1.3.19, falling a great distance to the surface is not the constant g. Rather, the acceleration a is inversely proportional to the square of the distance from the center of the Earth, a = k/r², where k is the constant of proportionality. Use the fact that at the surface of the Earth r = R and a = g to determine k. If the positive direction is upward, use Newton’s second law and his universal law of gravitation to find a differential equation for the distance r.
    

    

24. Suppose a hole is drilled through the center of the Earth and a bowling ball of mass m is dropped into the hole, as shown in FIGURE 1.3.20. Construct a mathematical model that describes the motion of the ball. At time t let r denote the distance from the center of the Earth to the mass m, M denote the mass of the Earth, M_r denote the mass of that portion of the Earth within a sphere of radius r, and δ denote the constant density of the Earth.
    

    

Additional Mathematical Models

25. Learning Theory In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t. Determine a differential equation for the amount A(t).
26. Forgetfulness In Problem 25, assume that the rate at which material is forgotten is proportional to the amount memorized in time t. Determine a differential equation for A(t) when forgetfulness is taken into account.
27. Infusion of a Drug A drug is infused into a patient’s bloodstream at a constant rate of r grams per second. Simultaneously, the drug is removed at a rate proportional to the amount x(t) of the drug present at time t. Determine a differential equation governing the amount x(t).
28. Tractrix A motorboat starts at the origin and moves in the direction of the positive x-axis, pulling a waterskier along a curve C called a tractrix. See FIGURE 1.3.21. The waterskier, initially located on the y-axis at the point (0, s), is pulled by keeping a rope of constant length s, which is kept taut throughout the motion. At time t > 0 the waterskier is at the point P(x, y). Find the differential equation of the path of motion C.
    

    

29. Reflecting Surface Assume that when the plane curve C shown in FIGURE 1.3.22 is revolved about the x-axis it generates a surface of revolution with the property that all light rays L parallel to the x-axis striking the surface are reflected to a single point O (the origin). Use the fact that the angle of incidence is equal to the angle of reflection to determine a differential equation that describes the shape of the curve C. Such a curve C is important in applications ranging from construction of telescopes to satellite antennas, automobile headlights, and solar collectors. [Hint: Inspection of the figure shows that we can write = 2θ. Why? Now use an appropriate trigonometric identity.]
    

    

    
    

    

Discussion Problems

30. Reread Problem 53 in Exercises 1.1 and then give an explicit solution P(t) for equation (1). Find a one-parameter family of solutions of (1).
31. Reread the sentence following equation (3) and assume that T_m is a positive constant. Discuss why we would expect k < 0 in (3) in both cases of cooling and warming. You might start by interpreting, say, T(t) > T_m in a graphical manner.
32. Reread the discussion leading up to equation (8). If we assume that initially the tank holds, say, 50 lb of salt, it stands to reason that since salt is being added to the tank continuously for t > 0, that A(t) should be an increasing function. Discuss how you might determine from the DE, without actually solving it, the number of pounds of salt in the tank after a long period of time.
33. Population Model The differential equation dP/dt = (k cos t)P, where k is a positive constant, is a model of human population P(t) of a certain community. Discuss an interpretation for the solution of this equation; in other words, what kind of population do you think the differential equation describes?
34. Rotating Fluid As shown in FIGURE 1.3.23(a), a right-circular cylinder partially filled with fluid is rotated with a constant angular velocity ω about a vertical y-axis through its center. The rotating fluid is a surface of revolution S. To identify S, we first establish a coordinate system consisting of a vertical plane determined by the y-axis and an x-axis drawn perpendicular to the y-axis such that the point of intersection of the axes (the origin) is located at the lowest point on the surface S. We then seek a function y = f(x), which represents the curve C of intersection of the surface S and the vertical coordinate plane. Let the point P(x, y) denote the position of a particle of the rotating fluid of mass m in the coordinate plane. See Figure 1.3.23(b).
    
    1. At P, there is a reaction force of magnitude F due to the other particles of the fluid, which is normal to the surface S. By Newton’s second law the magnitude of the net force acting on the particle is mω²x. What is this force? Use Figure 1.3.23(b) to discuss the nature and origin of the equations
       

       F cos θ = mg, F sin θ = mω²x.

    2. Use part (a) to find a first-order differential equation that defines the function y = f(x).
       

       

35. Falling Body In Problem 23, suppose r = R + s, where s is the distance from the surface of the Earth to the falling body. What does the differential equation obtained in Problem 23 become when s is very small compared to R?
36. Raindrops Keep Falling In meteorology, the term virga refers to falling raindrops or ice particles that evaporate before they reach the ground. Assume that a typical raindrop is spherical in shape. Starting at some time, which we can designate as t = 0, the raindrop of radius r₀ falls from rest from a cloud and begins to evaporate.
    
    1. If it is assumed that a raindrop evaporates in such a manner that its shape remains spherical, then it also makes sense to assume that the rate at which the raindrop evaporates—that is, the rate at which it loses mass—is proportional to its surface area. Show that this latter assumption implies that the rate at which the radius r of the raindrop decreases is a constant. Find r(t). [Hint: See Problem 63 in Exercises 1.1.]
    2. If the positive direction is downward, construct a mathematical model for the velocity v of the falling raindrop at time t. Ignore air resistance. [Hint: Use the form of Newton’s second law as given in (17).]
37. Let It Snow The “snowplow problem” is a classic and appears in many differential equations texts but was probably made famous by Ralph Palmer Agnew:
    

    One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour. What time did it start snowing?

    If possible, find the text Differential Equations, Ralph Palmer Agnew, McGraw-Hill, and then discuss the construction and solution of the mathematical model.

    

38. Reread this section and classify each mathematical model as linear or nonlinear.
39. Population Dynamics Suppose that P′(t) = 0.15 P(t) represents a mathematical model for the growth of a certain cell culture, where P(t) is the size of the culture (measured in millions of cells) at time t (measured in hours). How fast is the culture growing at the time t when the size of the culture reaches 2 million cells?
40. Radioactive Decay Suppose that
    

    A′(t) = −0.0004332 A(t)

    represents a mathematical model for the decay of radium-226, where A(t) is the amount of radium (measured in grams) remaining at time t (measured in years). How much of the radium sample remains at time t when the sample is decaying at a rate of 0.002 grams per year?

*If two quantities u and v are proportional, we write u ∝ v. This means one quantity is a constant multiple of the other: u = kv.

*Don’t confuse these symbols with R_in and R_out, which are input and output rates of salt.

*Note that when m is constant, this is the same as F = ma.