Explanation Calculation: Consider the differential equation of the given form. d y d x = f ( y ) y ( x ) is the solution of the autonomous differential equation and the equation in a sub-region R is bounded by two consecutive critical points c 1 < c 2 . Assume f ( y ) and f ′ ( y ) for the differential equation are continuous. The point y = c 1 within R such that f ( c 1 ) = 0 , if f ( y ) changes sign within a sub region R of the phase portrait. So, there will be an equilibrium solution y = c 1 within sub-region R . lim x → ∞ y ( x ) = c 2 lim x → ∞ y ′ ( x ) = 0 If this new critical point is between it will be a contradiction of consecutive critical points. A solution y(x) of d y d x = f ( y ) cannot have relative extrema and hence must be monotone. Since, y ′ ( x ) = f ( y ) > 0 , y ( x ) is monotone increasing, and since y ( x ) is bounded above by c 2

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 2.1, Problem 34E

To determine

The reason for the number L<c2 cannot exist for the given limit.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.1, Problem 33E

Chapter 2.1, Problem 35E

Students have asked these similar questions

4. Assume that a risk-free money market account is added to the market described in Q3. The continuously compounded rate of return on the money market account is log (1.1). (i) For each given μ, use Lagrange multipliers to determine the proportions (as a function of μ) of wealth invested in the three assets available for the minimum variance portfolio with expected return μ. (ii) Determine the market portfolio in this market and calculate its Sharp ratio.

3. A market consists of two risky assets with rates of return R₁ and R2 and no risk-free asset. From market data the following have been estimated: ER₁ = 0.25, ER2 = 0.05, Var R₁ = 0.01, Var R2 = 0.04 and the correlation between R1 and R2 is p = -0.75. (i) Given that an investor is targeting a total expected return of μ = 0.2. What portfolio weights should they choose to meet this goal with minimum portfolio variance? Correct all your calculations up to 4 decimal points. (ii) Determine the global minimum-variance portfolio and the expected return and variance of return of this portfolio (4 d.p.). (iii) Sketch the minimum-variance frontier in the μ-σ² plane and indicate the efficient frontier. (iv) Without further calculation, explain how the minimum variance of the investor's portfolio return will change if the two risky assets were independent.

2. A landlord is about to write a rental contract for a tenant which lasts T months. The landlord first decides the length T > 0 (need not be an integer) of the contract, the tenant then signs it and pays an initial handling fee of £100 before moving in. The landlord collects the total amount of rent erT at the end of the contract at a continuously compounded rate r> 0, but the contract stipulates that the tenant may leave before T, in which case the landlord only collects the total rent up until the tenant's departure time 7. Assume that 7 is exponentially distributed with rate > 0, λ‡r. (i) Calculate the expected total payment EW the landlord will receive in terms of T. (ii) Assume that the landlord has logarithmic utility U(w) = log(w - 100) and decides that the rental rate r should depend on the contract length T by r(T) = λ √T 1 For each given λ, what T (as a function of X) should the landlord choose so as to maximise their expected utility? Justify your answer. Hint. It might be…

Chapter 2 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - dydx=1xy (a) y(0) = 0 (b) y(1) = 0 (c) y(2) = 2...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The reason for the number L < c 2 cannot exist for the given limit.

Solution Summary: The author explains that the number Lc_2 cannot exist for the given limit.

Videos

The reason for the number L<c2 cannot exist for the given limit.

Want to see the full answer?

Chapter 2 Solutions

The reason for the number L &lt; c 2 cannot exist for the given limit.

Solution Summary: The author explains that the number Lc_2 cannot exist for the given limit.

Videos

The reason for the number L<c2 cannot exist for the given limit.

Want to see the full answer?

Chapter 2 Solutions

The reason for the number L < c 2 cannot exist for the given limit.