5.4.1 Use separation of variables to solve the following
autonomous differential equations. Check your answers by
differentiating. ...
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5.4.2
Use separation of variables to solve the following autonomous
differential equations. Check your answers by differentiating. ...
Get solution
5.4.3
Use separation of variables to solve the following autonomous
differential equations. Check your answers by differentiating. ...
Get solution
5.4.4
Use separation of variables to solve the following autonomous
differential equations. Check your answers by differentiating. ...
Get solution
5.4.5
Use separation of variables to solve the following autonomous
differential equations. Check your answers by differentiating. ...
Get solution
5.4.6
Use separation of variables to solve the following autonomous
differential equations. Check your answers by differentiating. ...
Get solution
5.4.7
Use separation of variables to solve the following pure-time
differential equations. Check your answers by differentiating. ...
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5.4.8
Use separation of variables to solve the following pure-time
differential equations. Check your answers by differentiating. ...
Get solution
5.4.9
Use separation of variables to solve the following pure-time
differential equations. Check your answers by differentiating. ...
Get solution
5.4.10
Use separation of variables to solve the following pure-time
differential equations. Check your answers by differentiating. ...
Get solution
5.4.11 Find the solution of the differential equation ...in the following cases. At what time does it approach infinity? Sketch a graph. p = 2 (as in the text) and b(0) = 100.
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5.4.12 Find the solution of the differential equation ...in the following cases. At what time does it approach infinity? Sketch a graph. p = 2 and b(0) = 0.1.
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5.4.13 Find the solution of the differential equation ...in the following cases. At what time does it approach infinity? Sketch a graph. p = 1.1 and b(0) = 100.
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5.4.14 Find the solution of the differential equation ...in the following cases. At what time does it approach infinity? Sketch a graph. p = 1.1 and b(0) = 0.1.
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5.4.15 The following autonomous differential equations can be solved except for the step of finding x in terms of t. Use the steps to figure out how the solution behaves.
a. Solve the equation with separation of variables.
b. It is impossible to algebraically solve for x in terms of t. However, you can still find the arbitrary constant. Find it.
c. Although you cannot find x as a function of t , you can find t as a function of x. Graph this function for 1 ≤ x ≤ 10.
d. Sketch the solution for x as a function of t. The autonomous differential equation ... with x(0)=1. This describes a population with per capita production that decreases like ....
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5.4.16 The following autonomous differential equations can be solved except for the step of finding x in terms of t. Use the steps to figure out how the solution behaves.
a. Solve the equation with separation of variables.
b. It is impossible to algebraically solve for x in terms of t. However, you can still find the arbitrary constant. Find it.
c. Although you cannot find x as a function of t , you can find t as a function of x. Graph this function for 1 ≤ x ≤ 10.
d. Sketch the solution for x as a function of t. The autonomous differential equation ...with x(0)=1.
This describes a population with per capita production that decreases
like .... Describe in words how the solution differs from that in
Exercise 15.
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5.4.17
Check the following results about the solution of the logistic
equation from Example 5.4.7. Show that the solution comes out the same
if N > K .
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5.4.18
Check the following results about the solution of the logistic
equation from Example 5.4.7. Differentiate to check that the solution
given in Example 5.4.7 solves the differential equation.
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5.4.19 Using
the method used to find the solution derived for Newton’s law of
cooling, find the solution of the chemical diffusion equation ...with
the following parameter values and initial conditions. Find the
concentration after 10 seconds. How long would it take for the
concentration to get halfway to the equilibrium value? ...
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5.4.20 Using
the method used to find the solution derived for Newton’s law of
cooling, find the solution of the chemical diffusion equation ...with
the following parameter values and initial conditions. Find the
concentration after 10 seconds. How long would it take for the
concentration to get halfway to the equilibrium value? ...
Get solution
5.4.21 Using
the method used to find the solution derived for Newton’s law of
cooling, find the solution of the chemical diffusion equation ...with
the following parameter values and initial conditions. Find the
concentration after 10 seconds. How long would it take for the
concentration to get halfway to the equilibrium value? ...
Get solution
5.4.22 Using
the method used to find the solution derived for Newton’s law of
cooling, find the solution of the chemical diffusion equation ...with
the following parameter values and initial conditions. Find the
concentration after 10 seconds. How long would it take for the
concentration to get halfway to the equilibrium value? ...
Get solution
5.4.23 Consider Torricelli’s law of draining ... (Section 5.2, Exercise 33) with the constant set to 2. Suppose the initial condition is y(0)= 4. Find the solution with separation of variables and graph the result. What really happens at time t = 2? And what happens after this time? How does this differ from the solution of the equation ...
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5.4.24 Consider Torricelli’s law of draining ... (Section 5.2, Exercise 33) with the constant set to 2. Suppose the initial condition is y(0)=
16. Find the solution with separation of variables and graph the
result. When does the solution reach 0? What would the depth be at this
time if draining followed the equation ...
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5.4.25 Suppose a population is growing at constant rate λ, but that individuals are harvested at a rate of h, following the differential equation ...= λb − h. For each of the following values of λ and h,
use separation of variables to find the solution, and compare graphs of
the solution with those found earlier from a phase-line diagram. λ = 2.0, h = 1000 (as in Section 5.2, Exercise 19).
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5.4.26 Suppose a population is growing at constant rate λ, but that individuals are harvested at a rate of h, following the differential equation ...= λb − h. For each of the following values of λ and h,
use separation of variables to find the solution, and compare graphs of
the solution with those found earlier from a phase-line diagram. λ = 0.5, h = 1000 (as in Section 5.2, Exercise 20).
Get solution
5.4.27 Use separation of variables to solve for C in the following models describing chemical diffusion, and find the solution starting from the initial condition C = Γ. The model in Section 5.1, Exercise 33. Reference Section 5.1, Exercise 33The
derivation of the movement of chemical assumed that chemical moved as
easily into the cell as out of it. If the membrane can act as a filter,
the rates at which chemical enters and leaves might differ, or might
depend on the concentration itsel
f. In each of the following cases, draw
a diagram illustrating the situation and write the associated
differential equation. Let C be the concentration inside the cell, Γ the concentration outside, and β the constant of proportionality relating the concentration and the rate. Suppose
that the constant of proportionality governing the rate at which
chemical enters the cell is three times as large as the constant
governing the rate at which it leaves. Would the concentration inside
the cell be increasing or decreasing if C = Γ? What would this mean for the cell?
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5.4.28 Use separation of variables to solve for C in the following models describing chemical diffusion, and find the solution starting from the initial condition C = Γ. The model in Section 5.1, Exercise 34.Reference Section 5.1, Exercise 34 Suppose
that the constant of proportionality governing the rate at which
chemical enters the cell is half as large as the constant governing the
rate at which it leaves. Would the concentration inside the cell be
increasing or decreasing if C = Γ? What would this mean for the cell?
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5.4.29 Separation
of variables can help to solve some nonautonomous differential
equations. For example, suppose that the per capita production rate is λ(t), a function of time, so that ... This equation can be separated into parts depending only on b and on t by dividing both sides of the equation by b and multiplying by dt. For the following functions λ(t), give an interpretation of the equation, and solve the equation with the initial condition b(0) = .... Sketch each solution. Check your answer by substituting into the differential equation. λ( t ) = t.
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5.4.30 Separation
of variables can help to solve some nonautonomous differential
equations. For example, suppose that the per capita production rate is λ(t), a function of time, so that ... This equation can be separated into parts depending only on b and on t by dividing both sides of the equation by b and multiplying by dt. For the following functions λ(t), give an interpretation of the equation, and solve the equation with the initial condition b(0) = .... Sketch each solution. Check your answer by substituting into the differential equation. ...
Get solution
5.4.31 Separation
of variables can help to solve some nonautonomous differential
equations. For example, suppose that the per capita production rate is λ(t), a function of time, so that ... This equation can be separated into parts depending only on b and on t by dividing both sides of the equation by b and multiplying by dt. For the following functions λ(t), give an interpretation of the equation, and solve the equation with the initial condition b(0) = .... Sketch each solution. Check your answer by substituting into the differential equation. λ( t ) = cos(t).
Get solution
5.4.32 Separation
of variables can help to solve some nonautonomous differential
equations. For example, suppose that the per capita production rate is λ(t), a function of time, so that ... This equation can be separated into parts depending only on b and on t by dividing both sides of the equation by b and multiplying by dt. For the following functions λ(t), give an interpretation of the equation, and solve the equation with the initial condition b(0) = .... Sketch each solution. Check your answer by substituting into the differential equation. ...
Get solution
5.4.33
Use integration by partial fractions to find solutions of the
following models. Describe the solution in words. The model from
Section 5.1, Exercise 29 with initial condition b(0) = 100.
Get solution
5.4.34
Use integration by partial fractions to find solutions of the
following models. Describe the solution in words. The model from
Section 5.1, Exercise 30 with initial condition b(0) = 10,000.
Get solution
5.4.35
Our
model of selection (Equation 5.1.4) looks much like the logistic
equation studied in Example 5.4.7, but can be extended to be
frequency-dependent when the fitness of one or both types depends on how
common it is. Find the solution of Equation 5.1.4 as a special case of
the logistic equation and compare with the solution in Equation 5.1.6.
Reference Equation 5.1.4 ... Reference Equation 5.1.6
...
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5.4.36
Our
model of selection (Equation 5.1.4) looks much like the logistic
equation studied in Example 5.4.7, but can be extended to be
frequency-dependent when the fitness of one or both types depends on how
common it is. Suppose that μ= 2 − 2p and λ= 1. Find the equilibria and their stability. What method would you use to try to find the solution after separating variables?
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5.4.37 There
are many important differential equations for which separation of
variables fails, but which can be solved with other techniques. An
important category involves Newton’s law of cooling when the ambient
temperature is changing. Consider, in particular, the case where .... Assume that the constant α is 1.0, so that the differential equation is ...=−H + A(t). It is impossible to separate variables in this equation. Create the new variable y = et H and find a differential equation for y.
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5.4.38 There
are many important differential equations for which separation of
variables fails, but which can be solved with other techniques. An
important category involves Newton’s law of cooling when the ambient
temperature is changing. Consider, in particular, the case where .... Assume that the constant α is 1.0, so that the differential equation is ...=−H + A(t).
It is impossible to separate variables in this equation. Identify the
type of differential equation, and solve it with the initial condition H(0) = 0.
Get solution
5.4.39 There
are many important differential equations for which separation of
variables fails, but which can be solved with other techniques. An
important category involves Newton’s law of cooling when the ambient
temperature is changing. Consider, in particular, the case where .... Assume that the constant α is 1.0, so that the differential equation is ...=−H + A(t). It is impossible to separate variables in this equation. Graph your solution and the ambient temperature when β is small, say β = 0.1. Describe the result.
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5.4.41 We have seen that solutions of the differential equation ... approach
infinity in a finite amount of time. We will try to stop it by
multiplying the rate of change by a decreasing function of t in the nonautonomous equation ...
a. Which of these functions decreases fastest and should best be able to stop b(t) from approaching infinity?
b. Have your computer solve the equation in each case with initial conditions ranging from b(0) = 0.1 to b(0) = 5.0. Which solutions approach infinity?
c. Use separation of variables to try to find the solution in each of these cases (have your computer help with the integral of ...Can you figure out when the solutions approach infinity? How well does this match your results from part b?
Get solution
5.4.41 We have seen that solutions of the differential equation ... approach
infinity in a finite amount of time. We will try to stop it by
multiplying the rate of change by a decreasing function of t in the nonautonomous equation ...
a. Which of these functions decreases fastest and should best be able to stop b(t) from approaching infinity?
b. Have your computer solve the equation in each case with initial conditions ranging from b(0) = 0.1 to b(0) = 5.0. Which solutions approach infinity?
c. Use separation of variables to try to find the solution in each of these cases (have your computer help with the integral of ...Can you figure out when the solutions approach infinity? How well does this match your results from part b?
Get solution