2 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-4.

In Problems 1–10, fill in the blanks or answer true/false.

  1. The DE y′ − ky = A, where k and A are constants, is autonomous. The critical point of the equation is a(n) (attractor or repeller) for k > 0 and a(n) (attractor or repeller) for k < 0.
  2. The initial-value problem has an infinite number of solutions for k = and no solution for k = .
  3. By inspection, two solutions of the differential equation are .
  4. Every autonomous DE is separable.
  5. The linear DE where are nonzero constants, possesses a constant solution.
  6. The first-order DE is not separable.
  7. If .
  8. The linear DE is also separable.
  9. The DE is a second-order equation.
  10. is a solution of the linear first-order differential equation .

In Problems 11 and 12, construct an autonomous first-order differential equation dy/dx = f(y) whose phase portrait is consistent with the given figure.

  1. A vertical line is labeled y. Two points, 1 and 3, are marked on the line, dividing it into 3 segments. Three arrows pointing up are marked in the segments: one below 1, another between 1 and 3, and another above 3.

    FIGURE 2.R.1 Phase portrait in Problem 11

  2. A vertical line is labeled y. Three points, 0, 2, and 4, are marked on the line, dividing it into 4 segments. Four arrows are marked. An arrow points upward above 4, points downward between 4 and 2, points downward between 2 and 0, and points up below 0.

    FIGURE 2.R.2 Phase portrait in Problem 12

  3. The number 0 is a critical point of the autonomous differential equation dx/dt = xn, where n is a positive integer. For what values of n is 0 asymptotically stable? Semi-stable? Unstable? Repeat for the equation dx/dt = −xn.
  4. Consider the differential equation

    ,    where   

    The function f(P) has one real zero, as shown in FIGURE 2.R.3. Without attempting to solve the differential equation, estimate the value of limt→ P(t).

    A graph. The horizontal axis is labeled P. The vertical axis is labeled f. A curve enters the second quadrant at approximately (negative 1, 5.5) and goes down and to the right through approximately (0, 3.4) and (0.8, 2). It then goes down and to the right more steeply through approximately (1.3, 0).

    FIGURE 2.R.3 Graph for Problem 14

  5. FIGURE 2.R.4 is a portion of the direction field of a differential equation dy/dx = f(x, y). By hand, sketch two different solution curves, one that is tangent to the lineal element shown in black and the other that is tangent to the lineal element shown in red.
    The image is divided into three horizontal sections. In the first section, the arrows point up and to the right, increasingly to the right, then down and to the right, increasingly to the right, then up and to the right. In the second section, the arrows point down and to the right, increasingly to the right, up and to the right, increasingly to the right, then down and to the right. A black arrow in the early part of the middle section points down and to the right. A red arrow in the middle part of the middle section points up and to the right. In the third section, the arrows point up and to the right, increasingly to the right, then down and to the right, increasingly to the right, then up and to the right.

    FIGURE 2.R.4 Direction field for Problem 15

  6. Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more than one kind. Do not solve.

    (a)

    (b)

    (c)

    (d)

    (e)

    (f)

    (g)

    (h)

    (i)

    (j)

    (k)

    (l)

    (m)

    (n)

In Problems 17–24, solve the given differential equation.

  8. (2r2 cos θ sin θ + r cos θ) dθ + (4r + sin θ − 2r cos2θ) dr = 0

In Problems 25–28, express the solution of the given initial-value problem in terms of an integral-defined function.

In Problems 29 and 30, solve the given initial-value problem.

In Problems 31 and 32, solve the given initial-value problem and give the largest interval I on which the solution is defined.

  1. ,
  2. ,
  3. (a) Without solving, explain why the initial-value problem

    ,

    has no solution for y0 < 0.

    (b) Solve the initial-value problem in part (a) for y0 > 0 and find the largest interval I on which the solution is defined.

  4. (a) Find an implicit solution of the initial-value problem

    ,

    (b) Find an explicit solution of the problem in part (a) and give the largest interval I over which the solution is defined. A graphing utility may be helpful here.

  5. Graphs of some members of a family of solutions for a first-order differential equation dy/dx = f(x, y) are shown in FIGURE 2.R.5. The graph of an implicit solution G(x, y) = 0 that passes through the points (1, −1) and (−1, 3) is shown in red. With colored pencils, trace out the solution curves of the solutions y = y1(x) and y = y2(x) defined by the implicit solution such that y1(1) = −1 and y2(−1) = 3. Estimate the interval I on which each solution is defined.
    A family of four curves is graphed. The first three curves consist of two parts. Curve 1, part 1. The curve enters the second quadrant, and goes down and to the right through approximately (negative 2, 4) to (negative 1, 0). It then goes down and to the right through approximately (negative 2, negative 4). Curve 1, part 2. An ellipse is graphed, passing through the approximate points (2.5, 0), (3, 1), (3.5, 0), and (3, negative 1). Curve 2, part 1. The curve enters the second quadrant, and goes down and to the right through approximately (negative 1, 2) to (negative 0.5, 0). It then goes down and to the right through approximately (negative 1, negative 2). Curve 2, part 2. A closed shape is graphed passing through the approximate points (1.5, 0), (3, 2.5), (4.2, 0), and (3, negative 2.5). Curve 3, part 1. The curve enters the second quadrant, and goes down and to the right through approximately (negative 1, 3) to (0, 0). It then goes down and to the right through approximately (negative 1, negative 3). Curve 3, part 2. A closed shape is graphed passing through the approximate points (0.8, 0), (3, 3), (4.3, 0) and (3, negative 3). The fourth curve enters the second quadrant and goes down and to the right through approximately (negative 1, 4), intersects the y axis at approximately (0, 1.8), and reaches a low point at approximately (0.5, 1.7). It then goes up and to the right to a high point at (3, 3.5), and goes down and to the right to approximately (4.4, 0). The curve then goes down and to the left to a low point at approximately (3, negative 3.5), and up and to the left to a high point at approximately (0.5, negative 1.7). The curve goes down and to the left, intersecting the y axis at approximately (0, negative 1.8), and continues through approximately (negative 1, negative 4).

    FIGURE 2.R.5 Graph for Problem 35

  6. Use Euler’s method with step size h = 0.1 to approximate y(1.2) where y(x) is a solution of the initial-value problem y′ = 1 + y(1) = 9.
  7. In March 1976, the world population reached 4 billion. A popular news magazine predicted that with an average yearly growth rate of 1.8%, the world population would be 8 billion in 45 years. How does this value compare with that predicted by the model that says the rate of increase is proportional to the population at any time t?
  8. Iodine-131 is a radioactive liquid used in the treatment of cancer of the thyroid. After one day in storage, analysis shows that initial amount of iodine-131 in a sample has decreased by 8.3%.

    (a) Find the amount of iodine-131 remaining in the sample after 8 days.

    (b) What is the significance of your answer in part (a)?

  9. Ötzi the Iceman In 1991 hikers found a well-preserved body of a man partially frozen in a glacier in the Ötztal Alps on the border between Austria and Italy. From the cuts and bruises found on his body, the trauma caused by a blow to his head, and an arrowhead found in his left shoulder, it is likely that Ötzi the Iceman—as he came to be called—was murdered. Through carbon-dating techniques, it was found that the body of Ötzi the Iceman contained 53% as much C-14 as found in a living person. Assume that the Iceman was carbon dated in 1991.

    (a) Using the Cambridge half-life of C-14, give an educated guess as to the date of his death.

    (b) Then use the technique illustrated in Example 3 of Section 2.7 to calculate the approximate date of his death.

    A photo of the remains of a human body.

    The Iceman in Problem 39

    © dpa/Corbis

  10. Air containing 0.06% carbon dioxide is pumped into a room whose volume is 8000 ft3. The air is pumped in at a rate of 2000 ft3/min, and the circulated air is then pumped out at the same rate. If there is an initial concentration of 0.2% carbon dioxide, determine the subsequent amount in the room at any time. What is the concentration at 10 minutes? What is the steady-state, or equilibrium, concentration of carbon dioxide?
  11. Solve the differential equation

    of the tractrix. See Problem 28 in Exercises 1.3. Assume that the initial point on the y-axis is (0, 10) and that the length of the rope is x = 10 ft.

  12. Suppose a cell is suspended in a solution containing a solute of constant concentration Cs. Suppose further that the cell has constant volume V and that the area of its permeable membrane is the constant A. By Fick’s law of diffusion, introduced in 1855 by the German physician/physiologist Adolf Eugen Fick (1829–1901), the rate of change of its mass m is directly proportional to the area A and the difference Cs − C(t), where C(t) is the concentration of the solute inside the cell at any time t. Find C(t) if m = V · C(t) and C(0) = C0. See FIGURE 2.R.6.
    A cell with a concentration of C of t is suspended in a solution. Molecules of solute diffuse through the cell membrane. The concentration of solute is C subscript s in the solution, and C of t in the cell.

    FIGURE 2.R.6 Cell in Problem 42

  13. Suppose that as a body cools, the temperature of the surrounding medium increases because it completely absorbs the heat being lost by the body. Let T(t) and Tm(t) be the temperatures of the body and the medium at time t, respectively. If the initial temperature of the body is T1 and the initial temperature of the medium is T2, then it can be shown in this case that Newton’s law of cooling is dT/dt = k(T − Tm), k < 0, where Tm = T2 + B(T1 − T), B > 0 is a constant.

    (a) The foregoing DE is autonomous. Use the phase portrait concept of Section 2.1 to determine the limiting value of the temperature T(t) as t → . What is the limiting value of Tm(t) as t → ?

    (b) Verify your answers in part (a) by actually solving the differential equation.

    (c) Discuss a physical interpretation of your answers in part (a).

  14. According to Stefan’s law of radiation, the absolute temperature T of a body cooling in a medium at constant temperature Tm is given by

    where k is a constant. Josef Stefan (1835–1893), a mathematical physicist born in the Austrian Empire, published his law of radiation in 1879. Stefan’s law can be used over a greater temperature range than Newton’s law of cooling.

    (a) Solve the differential equation.

    (b) Show that when T − Tm is small compared to Tm then Newton’s law of cooling approximates Stefan’s law. [Hint: Think binomial series of the right-hand side of the DE.]

  15. Suppose an RC-series circuit has a variable resistor. If the resistance at time t is defined by , where and are known positive constants, then the differential equation in (10) of Section 2.7 becomes

    where C is a constant. If and , where and are constants, then show that

  16. A classical problem in the calculus of variations is to find the shape of a curve 𝒞 such that a bead, under the influence of gravity, will slide from point A(0, 0) to point B(x1, y1) in the least time. See FIGURE 2.R.7. It can be shown that a nonlinear differential equation for the shape y(x) of the path is y[1 + (y′)2] = k, where k is a constant. First solve for dx in terms of y and dy, and then use the substitution y = k sin2θ to obtain a parametric form of the solution. The curve 𝒞 turns out to be a cycloid.
    The graph consists of a curve. A curve C begins at point A (0, 0) and goes down and to the right with decreasing steepness to a point B (x subscript one, y subscript one) in the fourth quadrant. A point on the curve represents a bead. An arrow pointing down from the bead is labeled m g.

    FIGURE 2.R.7 Sliding bead in Problem 46

    The clepsydra, or water clock, was a device used by the ancient Egyptians, Greeks, Romans, and Chinese to measure the passage of time by observing the change in the height of water that was permitted to flow out of a small hole in the bottom of a container or tank. In Problems 47–50, use the differential equation (see Problems 15–18 in Exercises 2.8)

    as a model for the height h of water in a tank at time t. Assume in each of these problems that h(0) = 2 ft corresponds to water filled to the top of the tank, the hole in the bottom is circular with radius in., g = 32 ft/s2, and c = 0.6.

  17. Suppose that a tank is made of glass and has the shape of a right-circular cylinder of radius 1 ft. Find the height h(t) of the water.
  18. For the tank in Problem 47, how far up from its bottom should a mark be made on its side, as shown in FIGURE 2.R.8, that corresponds to the passage of 1 hour? Continue and determine where to place the marks corresponding to the passage of 2 h, 3 h, . . ., 12 h. Explain why these marks are not evenly spaced.
    The clepsydra consists of a suspended cylindrical tank filled with water. The tank has a radius of one unit and a height of two units. The tank has a hole at the bottom. Horizontal lines on the tank indicate water level after one hour and two hours.

    FIGURE 2.R.8 Clepsydra in Problem 48

  19. Suppose that the glass tank has the shape of a cone with circular cross sections as shown in FIGURE 2.R.9. Can this water clock measure 12 time intervals of length equal to 1 hour? Explain using sound mathematics.
    The clepsydra consists of a suspended inverted conical tank filled with water. The tank has a radius of one unit and a height of two units. The tank has a hole at its vertex.

    FIGURE 2.R.9 Clepsydra in Problem 49

  20. Suppose that r = f(h) defines the shape of a water clock for which the time marks are equally spaced. Use the above differential equation to find f(h) and sketch a typical graph of h as a function of r. Assume that the cross-sectional area Ah of the hole is constant. [Hint: In this situation, dh/dt = −a, where a > 0 is a constant.]
  21. A model for the populations of two interacting species of animals is

    Solve for x and y in terms of t.

  22. Initially, two large tanks A and B each hold 100 gallons of brine. The well-stirred liquid is pumped between the tanks as shown in FIGURE 2.R.10. 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.
    Two tanks measuring 100 gallons each are labeled A and B. Five pipes are connected to the tanks and the rate of flow is as follows. Into A: 2 pounds per gallon, 7 gallons per minute. From B to A: mixture at one gallon per minute. From A to B: mixture at five gallons per minute. Out of A: mixture at 3 gallons per minute. Out of B: mixture at four gallons per minute.

    FIGURE 2.R.10 Mixing tanks in Problem 52

  23. It is estimated that the ecosystem of Yellowstone National Park can sustain a grey wolf population of 450. An initial population in 1997 was 40 grey wolves, and it was subsequently determined that the population grew to 95 wolves after 15 years. How many wolves does the mathematical model

    predict there will be in the park 30 years after their introduction?

  24. (a) Use a graphing utility to graph the wolf population P(t) found in Problem 53.

    (b) Use the solution P(t) in Problem 53 to find P(t).

    (c) Show that the differential equation in Problem 53 is a special case of Gompertz’s equation ((7) in Section 2.8).

    When all the curves in a family G(x, y, c1) = 0 intersect orthogonally all the curves in another family H(x, y, c2) = 0, the families are said to be orthogonal trajectories of each other. See FIGURE 2.R.11. If dy/dx = f(x, y) is the differential equation of one family, then the differential equation for the orthogonal trajectories of this family is dy/dx = −1/f(x, y). In Problems 55–58, find the differential equation of the given family. Find the orthogonal trajectories of this family. Use a graphing utility to graph both families on the same set of coordinate axes.

    Two sets of curves are graphed. Three curves representing G of x y, and c subscript one equals zero, go up and to the right with increasing steepness. Three curves representing H of x y, and c subscript two equals zero, goes down and to the right with increasing steepness. The curves intersect. The tangents to the curves are marked at one point of intersection. The tangents are perpendicular to each other.

    FIGURE 2.R.11 Orthogonal trajectories

  28. When the orthogonal trajectories of a one-parameter family of curves are itself, then this family is said to be self-orthogonal. In Problems 59 and 60, use the given method to show that the family of confocal parabolas is self-orthogonal.

  29. Suppose the family is defined in the piecewise manner:

    (parabolas opening to the right),

    (parabolas opening to the left).

    Find the points of intersection of the curves defined by these two equations. Then show that the tangent lines are perpendicular at the points of intersection.

  30. Show that the differential equation that defines the family and the differential equation that defines the orthogonal trajectories of the given family are the same.

Contributed Problem

Rick Wicklin, PhD
Senior Researcher in Computational Statistics, SAS Institute Inc.

  1. Invasion of the Marine Toads*     In 1935, the poisonous American marine toad (Bufo marinus) was introduced, against the advice of ecologists, into some of the coastal sugar cane districts in Queensland, Australia, as a means of controlling sugar cane beetles. Due to lack of natural predators and the existence of an abundant food supply, the toad population grew and spread into regions far from the original districts. The survey data given in the accompanying table indicate how the toads expanded their territorial bounds within a 40-year period. Our goal in this problem is to find a population model of the form but we want to construct the model that best fits the given data. Note that the data are not given as number of toads at 5-year intervals since this kind of information would be virtually impossible to obtain.
    A photo of a marine toad.

    Marine toad (Bufo marinus)

    © Ryan M. Bolton/ShutterStock, Inc.

    Year Area Occupied
    1939   32,800
    1944   55,800
    1949   73,600
    1954 138,000
    1959 202,000
    1964 257,000
    1969 301,000
    1974 584,000
    1. For ease of computation, let’s assume that, on the average, there is one toad per square kilometer. We will also count the toads in units of thousands and measure time in years with corresponding to 1939. One way to model the data in the table is to use the initial condition and to search for a value of k so that the graph of appears to fit the data points. Experiment, using a graphic calculator or a CAS, by varying the birth rate k until the graph of appears to fit the data well over the time period

      Alternatively, it is also possible to solve analytically for a value of k that will guarantee that the curve passes through exactly two of the data points. Find a value of k so that Find a different value of k so that

    2. In practice, a mathematical model rarely passes through every experimentally obtained data point, and so statistical methods must be used to find values of the model’s parameters that best fit experimental data. Specifically, we will use linear regression to find a value of k that describes the given data points:
      • Use the table to obtain a new data set of the form where is the given population at the times . . . .
      • Most graphic calculators have a built-in routine to find the line of least squares that fits this data. The routine gives an equation of the form where m and b are, respectively, the slope and intercept corresponding to the line of best fit. (Most calculators also give the value of the correlation coefficient that indicates how well the data are approximated by a line; a correlation coefficient of 1 or means perfect correlation. A correlation coefficient near 0 may mean that the data do not appear to be fit by an exponential model.)
      • Solving gives or P(t) = ebemt. Matching the last form with we see that is an approximate initial population, and m is the value of the birth rate that best fits the given data.
    3. So far you have produced four different values of the birth rate k. Do your four values of k agree closely with each other? Should they? Which of the four values do you think is the best model for the growth of the toad population during the years for which we have data? Use this birth rate to predict the toad’s range in the year 2039. Given that the area of Australia is 7,619,000 km2, how confident are you of this prediction? Explain your reasoning.
  2. Invasion of the Marine Toads—Continued     In part (a) of Problem 61, we made the assumption that there was an average of one toad per square kilometer. But suppose we are wrong and there were actually an average of two toads per square kilometer. As before, solve analytically for a value of k that will guarantee that the curve passes through exactly two of the data points. In particular, if we now assume that find a value of k so that and a different value of k so that How do these values of k compare with the values you found previously? What does this tell us? Discuss the importance of knowing the exact average density of the toad population.

 

*This problem is based on the article “Teaching Differential Equations with a Dynamical Systems Viewpoint” by Paul Blanchard, The College Mathematics Journal 25 (1994), 385–395.