In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = 0.9yn
Read more
Table of Contents
1.1
Some Basic Mathematical Models; Direction Fields
1.2
Solutions of Some Differential Equations
1.3
Classification of Differential Equations
2
First Order
Differential Equations
2.1
Linear Equations; Method of Integrating Factors
2.2
Separable Equations
2.3
Modeling with First Order Equations
2.4
Differences Between Linear and Nonlinear Equations
2.5
Autonomous Equations and Population Dynamics
2.6
Exact Equations and Integrating Factors
2.7
Numerical Approximations: Eulers Method
2.8
The Existence and Uniqueness Theorem
2.9
First Order Difference Equations
3.1
Homogeneous Equations with Constant Coefficients
3.2
Solutions of Linear Homogeneous Equations; the Wronskian
3.3
Complex Roots of the Characteristic Equation
3.4
Repeated Roots; Reduction of Order
3.5
Nonhomogeneous Equations; Method of Undetermined Coefficients
3.6
Variation of Parameters
3.7
Mechanical and Electrical Vibrations
3.8
Forced Vibrations
4.1
General Theory of nth Order Linear Equations
4.2
Homogeneous Equations with Constant Coefficients
4.3
The Method of Undetermined Coefficients
4.4
The Method of Variation of Parameters
5.1
Elementary Differential Equations, 10th Edition 9780470458327 William E. Boyce / Richard C. DiPrima
5.2
Series Solutions Near an Ordinary Point, Part I
5.3
Series Solutions Near an Ordinary Point, Part II
5.4
Euler Equations; Regular Singular Points
5.5
Series Solutions Near a Regular Singular Point, Part I
5.6
Series Solutions Near a Regular Singular Point, Part II
5.7
Bessels Equation
6.1
Definition of the Laplace Transform
6.2
Solution of Initial Value Problems
6.3
Step Functions
6.4
Differential Equations with Discontinuous Forcing Functions
6.5
Impulse Functions
6.6
The Convolution Integral
7.1
Introduction
7.2
Review of Matrices
7.3
Systems of Linear Algebraic Equations; Linear Independence,
Eigenvalues, Eigenvectors
7.4
Elementary Differential Equations, 10th Edition 9780470458327 William E. Boyce / Richard C. DiPrima
7.5
Homogeneous Linear Systems with Constant Coefficients
7.6
Complex Eigenvalues
7.7
Fundamental Matrices
7.8
Repeated Eigenvalues
7.9
Nonhomogeneous Linear Systems
8.1
The Euler or Tangent Line Method
8.2
Improvements on the Euler Method
8.3
The RungeKutta Method
8.4
Multistep Methods
8.5
Systems of First Order Equations
8.6
More on Errors; Stability
9.1
The Phase Plane: Linear Systems
9.2
Autonomous Systems and Stability
9.3
Locally Linear Systems
9.4
Competing Species
9.5
PredatorPrey Equations
9.6
Liapunovs Second Method
9.7
Periodic Solutions and Limit Cycles
9.8
Chaos and Strange Attractors: The Lorenz Equations
Textbook Solutions for Elementary Differential Equations
Chapter 2.9 Problem 19
Question
The Logistic Difference Equation. 14 through 19 deal with the difference equation(21), un+1 = un(1 un).Let k be the value of at which the solution of Eq. (21) changes from period 2k1 toperiod 2k. Thus, as noted in the text, 1 = 3, 2 = 3.449, and 3 = 3.544.(a) Using these values of 1, 2, and 3, or those you found in 18, calculate(2 1)/(3 2).(b) Let n = (n n1)/(n+1 n).It can be shown that n approaches a limit as n ,where = 4.6692 is known as the Feigenbaum22 number. Determine the percentagedifference between the limiting value and 2, as calculated in part (a).(c) Assume that 3 = and use this relation to estimate 4, the value of at which solutionsof period 16 appear.(d) By plotting or calculating solutions near the value of 4 found in part (c), try to detectthe appearance of a period 16 solution.(e) Observe thatn = 1 + (2 1) + (3 2) ++ (n n1).Assuming that (4 3) = (3 2)1,(5 4) = (3 2)2, and so forth, express nas a geometric sum. Then find the limit of n as n . This is an estimate of the value of at which the onset of chaos occurs in the solution of the logistic equation (21).
Solution
The first step in solving 2.9 problem number 19 trying to solve the problem we have to refer to the textbook question: The Logistic Difference Equation. 14 through 19 deal with the difference equation(21), un+1 = un(1 un).Let k be the value of at which the solution of Eq. (21) changes from period 2k1 toperiod 2k. Thus, as noted in the text, 1 = 3, 2 = 3.449, and 3 = 3.544.(a) Using these values of 1, 2, and 3, or those you found in 18, calculate(2 1)/(3 2).(b) Let n = (n n1)/(n+1 n).It can be shown that n approaches a limit as n ,where = 4.6692 is known as the Feigenbaum22 number. Determine the percentagedifference between the limiting value and 2, as calculated in part (a).(c) Assume that 3 = and use this relation to estimate 4, the value of at which solutionsof period 16 appear.(d) By plotting or calculating solutions near the value of 4 found in part (c), try to detectthe appearance of a period 16 solution.(e) Observe thatn = 1 + (2 1) + (3 2) ++ (n n1).Assuming that (4 3) = (3 2)1,(5 4) = (3 2)2, and so forth, express nas a geometric sum. Then find the limit of n as n . This is an estimate of the value of at which the onset of chaos occurs in the solution of the logistic equation (21).
From the textbook chapter First Order Difference Equations you will find a few key concepts needed to solve this.
Visible to paid subscribers only
Step 3 of 7)Visible to paid subscribers only
Subscribe to view the
full solution
full solution
Title
Elementary Differential Equations 10
Author
William E. Boyce, Richard C. DiPrima
ISBN
9780470458327
Get answer: The Logistic Difference Equation. 14 through 19 deal with the difference
Chapter 2.9 textbook questions
-
Chapter 2: Problem 1 Elementary Differential Equations 10
-
Chapter 2: Problem 2 Elementary Differential Equations 10
In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = n + 1 n + 2 yn
Read more -
Chapter 2: Problem 3 Elementary Differential Equations 10
In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = n + 3 n + 1 yn
Read more -
Chapter 2: Problem 4 Elementary Differential Equations 10
In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = (1)n+1yn
Read more -
Chapter 2: Problem 5 Elementary Differential Equations 10
In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = 0.5yn + 6
Read more -
Chapter 2: Problem 6 Elementary Differential Equations 10
In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = 0.5yn + 6
Read more -
Chapter 2: Problem 7 Elementary Differential Equations 10
Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance
Read more -
Chapter 2: Problem 8 Elementary Differential Equations 10
An investor deposits $1000 in an account paying interest at a rate of 8%, compounded monthly, and also makes additional deposits of $25 per month. Find the balance in the account after 3 years
Read more -
Chapter 2: Problem 9 Elementary Differential Equations 10
A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%.What monthly payment rate is required to pay off the loan in 3 years? Compare your result with that of Problem 9 in Section 2.3.
Read more -
Chapter 2: Problem 10 Elementary Differential Equations 10
A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period.What monthly payment is required if the interest rate is (a) 9%, (b) 10%, (c) 12%?
Read more -
Chapter 2: Problem 11 Elementary Differential Equations 10
A homebuyer takes out a mortgage of $100,000 with an interest rate of 9%.What monthly payment is required to pay off the loan in 30 years? In 20 years? What is the total amount paid during the term of the loan in each of these cases?
Read more -
Chapter 2: Problem 12 Elementary Differential Equations 10
If the interest rate on a 20-year mortgage is fixed at 10% and if a monthly payment of $1000 is the maximum that the buyer can afford, what is the maximum mortgage loan that can be made under these conditions?
Read more -
Chapter 2: Problem 13 Elementary Differential Equations 10
A homebuyer wishes to finance the purchase with a $95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed $900?
Read more -
Chapter 2: Problem 14 Elementary Differential Equations 10
The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). Carry out the details in the linear stability analysis of the equilibrium solution un = ( 1)/. That is, derive the difference equation (26) in the text for the perturbation vn.
Read more -
Chapter 2: Problem 15 Elementary Differential Equations 10
The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). (a) For = 3.2, plot or calculate the solution of the logistic equation (21) for several initial conditions, say, u0 = 0.2, 0.4, 0.6, and 0.8. Observe that in each case the solution approaches a steady oscillation between the same two values. This illustrates that the long-term behavior of the solution is independent of the initial conditions. (b) Make similar calculations and verify that the nature of the solution for large n is independent of the initial condition for other values of , such as 2.6, 2.8, and 3.4.
Read more -
Chapter 2: Problem 16 Elementary Differential Equations 10
The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). Assume that > 1 in Eq. (21). (a) Draw a qualitatively correct stairstep diagram and thereby show that if u0 < 0, then un as n . (b) In a similar way, determine what happens as n if u0 > 1.
Read more -
Chapter 2: Problem 17 Elementary Differential Equations 10
The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). The solutions of Eq. (21) change from convergent sequences to periodic oscillations of period 2 as the parameter passes through the value 3. To see more clearly how this happens, carry out the following calculations. (a) Plot or calculate the solution for = 2.9, 2.95, and 2.99, respectively, using an initial value u0 of your choice in the interval (0, 1). In each case estimate how many iterations are required for the solution to get very close to the limiting value. Use any convenient interpretation of what very close means in the preceding sentence. (b) Plot or calculate the solution for = 3.01, 3.05, and 3.1, respectively, using the same initial condition as in part (a). In each case estimate how many iterations are needed to reach a steady-state oscillation. Also find or estimate the two values in the steady-state oscillation.
Read more -
Chapter 2: Problem 18 Elementary Differential Equations 10
The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). By calculating or plotting the solution of Eq. (21) for different values of , estimate the value of at which the solution changes from an oscillation of period 2 to one of period 4. In the same way, estimate the value of at which the solution changes from period 4 to period 8.
Read more -
Chapter 2: Problem 19 Elementary Differential Equations 10
The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). Let k be the value of at which the solution of Eq. (21) changes from period 2k1 to period 2k. Thus, as noted in the text, 1 = 3, 2 = 3.449, and 3 = 3.544. (a) Using these values of 1, 2, and 3, or those you found in Problem 18, calculate (2 1)/(3 2). (b) Let n = (n n1)/(n+1 n).It can be shown that n approaches a limit as n , where = 4.6692 is known as the Feigenbaum22 number. Determine the percentage difference between the limiting value and 2, as calculated in part (a). (c) Assume that 3 = and use this relation to estimate 4, the value of at which solutions of period 16 appear. (d) By plotting or calculating solutions near the value of 4 found in part (c), try to detect the appearance of a period 16 solution. (e) Observe that n = 1 + (2 1) + (3 2) ++ (n n1). Assuming that (4 3) = (3 2)1,(5 4) = (3 2)2, and so forth, express n as a geometric sum. Then find the limit of n as n . This is an estimate of the value of at which the onset of chaos occurs in the solution of the logistic equation (21).
Read more