5.2.1 Find the equilibria of the following autonomous differential equations. ...
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5.2.2 Find the equilibria of the following autonomous differential equations. ...
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5.2.3 Find the equilibria of the following autonomous differential equations. ...
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5.2.4 Find the equilibria of the following autonomous differential equations. ...
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5.2.5 Find the equilibria of the following autonomous differential equations that include parameters. ...
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5.2.6 Find the equilibria of the following autonomous differential equations that include parameters. ...
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5.2.7 Find the equilibria of the following autonomous differential equations that include parameters. ...
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5.2.8 Find the equilibria of the following autonomous differential equations that include parameters. ...
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5.2.9 From the following graphs of the rate of change as a function of the state variable, draw the phase-line diagram. ...
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5.2.10 From the following graphs of the rate of change as a function of the state variable, draw the phase-line diagram. ...
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5.2.11 From the following phase-line diagrams, sketch a solution starting from the specified initial condition. ...
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5.2.12 From the following phase-line diagrams, sketch a solution starting from the specified initial condition. ...
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5.2.13 From the given phase-line diagram, sketch a possible graph of the rate of change of x as a function of x. The phase line in Exercise 11. Reference Exercise 11 ...
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5.2.14 From the given phase-line diagram, sketch a possible graph of the rate of change of x as a function of x. The phase line in Exercise 12. Reference Exercise 12 ...
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5.2.15 Graph
the rate of change as a function of the state variable and draw the
phase-line diagram for the following differential equations. ... (as in Exercise 1). Graph for −2 ≤ x ≤ 2.
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5.2.16 Graph
the rate of change as a function of the state variable and draw the
phase-line diagram for the following differential equations. ... (as in Exercise 2). Graph for −2 ≤ x ≤ 2.
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5.2.17 Graph
the rate of change as a function of the state variable and draw the
phase-line diagram for the following differential equations. ... (as in Exercise 3). Graph for −2 ≤ y ≤ 2.
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5.2.18 Graph
the rate of change as a function of the state variable and draw the
phase-line diagram for the following differential equations. ... (as in Exercise 4). Graph for 0 < z ≤ 1.
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5.2.19 Suppose a population is growing at constant rate λ, but that individuals are harvested at a rate of h. The differential equation describing such a population is ... For each of the following values of λ and h,
find the equilibrium, draw the phase-line diagram, and sketch one
solution with initial condition below the equilibrium and another with
initial condition above the equilibrium. Explain your result in words. λ = 2.0, h =1000.
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5.2.20 Suppose a population is growing at constant rate λ, but that individuals are harvested at a rate of h. The differential equation describing such a population is ... For each of the following values of λ and h,
find the equilibrium, draw the phase-line diagram, and sketch one
solution with initial condition below the equilibrium and another with
initial condition above the equilibrium. Explain your result in words. λ = 0.5, h = 1000.
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5.2.21 Find the equilibria, graph the rate of change ...as a function of b,
and draw a phase-line diagram for the following models describing
bacterial population growth. The model in Section 5.1, Exercise 27.
Check that your arrows are consistent with the behavior of b(t) at b = 10 and b = 1000.
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5.2.22 Find the equilibria, graph the rate of change ...as a function of b,
and draw a phase-line diagram for the following models describing
bacterial population growth. The model in Section 5.1, Exercise 28.
Check that your arrows are consistent with the behavior of b(t) at b = 1000 and b = 5000.
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5.2.23 Find the equilibria, graph the rate of change ...as a function of b,
and draw a phase-line diagram for the following models describing
bacterial population growth. The model in Section 5.1, Exercise 29.
Check that your arrows are consistent with the behavior of b(t) at b = 100 and b = 300.
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5.2.24 Find the equilibria, graph the rate of change ...as a function of b,
and draw a phase-line diagram for the following models describing
bacterial population growth. The model in Section 5.1, Exercise 30.
Check that your arrows are consistent with the behavior of b(t) at b = 1000 and b = 3000.
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5.2.25 Find the equilibria, graph the rate of change ...as a function of C,
and draw a phase-line diagram for the following models describing
chemical diffusion. The model in Section 5.1, Exercise 33. Check that
the direction arrow is consistent with the behavior of C(t) at C = Γ.
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5.2.26 Find the equilibria, graph the rate of change ...as a function of C,
and draw a phase-line diagram for the following models describing
chemical diffusion. The model in Section 5.1, Exercise 34. Check that
the direction arrow is consistent with the behavior of C(t) at C = Γ.
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5.2.27 Find the equilibria, graph the rate of change ...as a function of p,
and draw a phase-line diagram for the following models describing
selection. The model in Section 5.1, Exercise 37. What happens to a
solution starting from a small, but positive, value of p?
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5.2.28 Find the equilibria, graph the rate of change ...as a function of p,
and draw a phase-line diagram for the following models describing
selection. The model in Section 5.1, Exercise 38. What happens to a
solution starting from a small, but positive, value of p?
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5.2.29
Find
the equilibria and draw the phase-line diagram for the following
differential equations, in addition to answering the questions.
Suppose the population size of some species of organism follows the
model ... where N is
measured in hundreds. Why might this population behave as it does at
small values? This is another example of the Allee effect discussed in
Section 5.1, Exercise 29.
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5.2.30
Find
the equilibria and draw the phase-line diagram for the following
differential equations, in addition to answering the questions.
Suppose the population size of some species of organism follows the
model ... where N is measured in hundreds. What is the critical value below which this population is doomed to extinction (as in Exercise 29)?
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5.2.31
Find
the equilibria and draw the phase-line diagram for the following
differential equations, in addition to answering the questions. Suppose
the population size of some species of organism follows the model
... where v is speed, a is acceleration, and D is drag. Suppose that a = 9.8 ...and that D = 0.0032
per meter (values for a skydiver). Check that the units in the
differential equation are consistent. What does the equilibrium speed
mean?
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5.2.32 Find
the equilibria and draw the phase-line diagram for the following
differential equations, in addition to answering the questions. Consider
the same situation as in Exercise 31 but for a skydiver diving head
down with her arms against her sides and her toes pointed, thus
minimizing drag. The drag D is reduced to D = 0.00048 per meter. Find the equilibrium speed. How does it compare to the ordinary skydiver?
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5.2.33 Find
the equilibria and draw the phase-line diagram for the following
differential equations, in addition to answering the questions. According
to Torricelli’s law of draining, the rate that a fluid flows out of a
cylinder through a hole at the bottom is proportional to the square root
of the depth of the water. Let y represent the depth of water in centimeters. The differential equation is ... where .... Show that the units are consistent. Use your phase-line diagram to sketch solutions starting from y = 10.0 and y = 1.0.
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5.2.34 Find
the equilibria and draw the phase-line diagram for the following
differential equations, in addition to answering the questions. Write
a differential equation describing the depth of water in a cylinder
where water enters at a rate of 4.0 cm/s but drains out as in Exercise
33. Use your phase-line diagram to sketch solutions starting from y = 10.0 and y = 1.0.
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5.2.35
Find
the equilibria and draw the phase-line diagram for the following
differential equations, in addition to answering the questions. One of
the most important differential equations in chemistry uses the
Michaelis-Menton or Monod equation. Suppose S is the
concentration of a substrate that is being converted into a product.
Then ... describes how substrate is used. Set .... How does this equation differ from Torricelli’s law of draining (Exercise 33)?
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5.2.36
Find
the equilibria and draw the phase-line diagram for the following
differential equations, in addition to answering the questions. One of
the most important differential equations in chemistry uses the
Michaelis-Menton or Monod equation. Write a differential equation
describing the amount of substrate if substrate is added at rate R but
is converted into product as in Exercise 35. Find the equilibrium, and
draw the phase-plane diagram and a representative solution with R = 0.5 and R = 1.5. Can you explain your results?
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5.2.37
Small organisms like bacteria take in food at rates proportional to
their surface area but use energy at higher rates. Suppose that energy
is used at a rate proportional to the mass. In this case, ...
where V represents the volume in cubic centimeters and t is time measured in days. The first term says that surface area is proportional to volume to the 2/3 power. The constant ... gives the rate at which energy is taken in and has units of centimeters per day. ...
is rate at which energy is used and has units of per day. Check the
units. Find the equilibrium. What happens to the equilibrium as ... becomes smaller? Does this make sense? What happens to the equilibrium as ... becomes smaller? Does this make sense?
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5.2.38
Small organisms like bacteria take in food at rates proportional to
their surface area but use energy at higher rates. Suppose that
energy is used at a rate proportional to the mass to the 3/4 power (as
in Example 1.7.24). In this case, ... Find the units of ... if V is measured in ... and t is measured in days (they should look rather strange). Find the equilibrium. What happens to the equilibrium as ... becomes smaller? Does this make sense? What happens to the equilibrium as ... becomes smaller? Does this make sense?
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5.2.39 Consider the differential equation ... for various positive values of the parameter p starting from b(0) = 1.0. For which values of p does the solution approach the equilibrium at b =
0 most quickly? Plot the solution on a semilog graph. When does the
solution approach zero faster than an exponential function?
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5.2.40 In Exercise 53 we considered the equation ... Using
either the solution given in that problem or a computer system that can
solve the equation, show that the solution always approaches the
“equilibrium” at H = A(t). Use the parameter values α = 1.0 and values of T ranging from 0.1 to 10.0. For which value of T does the solution get closest to A(t)? Can you explain why?
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