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Chapter 5: Q26E (page 271)

Using the software, sketch the direction field in the phase-plane for the systemdxdt=y,dydt=-x-x3. From the sketch, conjecture whether all solutions of this system are bounded. Solve the related phase plane differential equation and confirm your conjecture.

Short Answer

Expert verified

The solutions of the system are bounded.

Step by step solution

01

Find a critical point

Here the equation is:

dxdt=ydydt=-x-x3

And

dydx=-x-x3y鈭�ydy=鈭�-x-x3dxy22=-x22-x44+c

02

Sketch the Directional field. 

03

Sketch for the solution.

Thus, the system is bounded.

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Most popular questions from this chapter

A house, for cooling purposes, consists of two zones: the attic area zone A and the living area zone B (see Figure 5.4). The living area is cooled by a 2 鈥� ton air conditioning unit that removes 24,000 Btu/hr. The heat capacity of zone B is12F鈭�per thousand Btu. The time constant for heat transfer between zone A and the outside is 2 hr, between zone B and the outside is 4 hr, and between the two zones is 4 hr. If the outside temperature stays at 100掳F, how warm does it eventually get in the attic zone A? (Heating and cooling buildings was treated in Section 3.3 on page 102.)

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Logistic Model.In Section 3.2 we discussed the logistic equation\(\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = A}}{{\bf{p}}_{\bf{1}}}{\bf{p - A}}{{\bf{p}}^{\bf{2}}}{\bf{,p(0) = }}{{\bf{p}}_{\bf{o}}}\)and its use in modeling population growth. A more general model might involve the equation\(\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = A}}{{\bf{p}}_{\bf{1}}}{\bf{p - A}}{{\bf{p}}^{\bf{r}}}{\bf{,p(0) = }}{{\bf{p}}_{\bf{o}}}\)where r>1. To see the effect of changing the parameter rin (25), take \({{\bf{p}}_{\bf{1}}}\)= 3, A= 1, and \({{\bf{p}}_{\bf{o}}}\)= 1. Then use a numerical scheme such as Runge鈥揔utta with h= 0.25 to approximate the solution to (25) on the interval\(0 \le {\bf{t}} \le 5\) for r= 1.5, 2, and 3What is the limiting population in each case? For r>1, determine a general formula for the limiting population.

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d2ydt2+y3=0.
See all solutions

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