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Chapter 5: Q25E (page 271)

Using the software, sketch the direction field in the phase-plane for the system dxdt=y,dydt=-x+x3From the sketch, conjecture whether the solution passing through each given point is periodic:

  1. (0.25, 0.25)
  2. (2, 2)
  3. (1, 0)

Short Answer

Expert verified
  1. The solution is periodic.
  2. The solution is non-periodic.
  3. The solution of the critical point (1,0) and periodic.

Step by step solution

01

Find the critical point

Here the equation is:

dxdt=ydydt=-x+x3

For critical points equate the system equal to zero.

y=0-x+x3=0x(-1+x2)=0x=0  or   (-1+x2)=0x=0,1

02

Sketch

03

Solution of a

The solution passing through the point (0.25,0.25) flows around (0,0) and this is periodic.

04

Result of part (b)

The solution passes through the point (2,2), y(t)→∞  as  t→∞and this is not periodic.

05

Find the solution (c)

The solution passing through the point (1,0) is a constant solution and this is periodic.

This is the required result.

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Most popular questions from this chapter

In Problems 29 and 30, determine the range of values (if any) of the parameter that will ensure all solutions x(t), and y(t) of the given system remain bounded as t→+∞.

dxdt=λx-y,dydt=3x+y

Rigid Body Nutation. Euler’s equations describe the motion of the principal-axis components of the angular velocity of a freely rotating rigid body (such as a space station), as seen by an observer rotating with the body (the astronauts, for example). This motion is called nutation. If the angular velocity components are denoted by x, y, and z, then an example of Euler’s equations is the three-dimensional autonomous system

\(\begin{array}{l}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = yz}}\\\frac{{{\bf{dy}}}}{{{\bf{dt}}}}{\bf{ = - 2xz}}\\\frac{{{\bf{dz}}}}{{{\bf{dt}}}}{\bf{ = xy}}\end{array}\)

The trajectory of a solution x(t),y(t), z(t) to these equations is the curve generated by the points (x(t), y(t), z(t) ) in xyz-phase space as t varies over an interval I.

(a) Show that each trajectory of this system lies on the surface of a (possibly degenerate) sphere centered at the origin (0, 0, 0).[Hint: Compute\(\frac{{\bf{d}}}{{{\bf{dt}}}}{\bf{(}}{{\bf{x}}^{\bf{2}}}{\bf{ + }}{{\bf{y}}^{\bf{2}}}{\bf{ + }}{{\bf{z}}^{\bf{2}}}{\bf{)}}\)What does this say about the magnitude of the angular velocity vector?

(b) Find all the critical points of the system, i.e., all points\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{y}}_{\bf{o}}}{\bf{,}}{{\bf{z}}_{\bf{o}}}{\bf{)}}\) such that \({\bf{x(t) = }}{{\bf{x}}_{\bf{o}}}{\bf{,y(t) = }}{{\bf{y}}_{\bf{o}}}{\bf{,z(t) = }}{{\bf{z}}_{\bf{o}}}\) is a solution. For such solutions, the angular velocity vector remains constant in the body system.

(c) Show that the trajectories of the system lie along the intersection of a sphere and an elliptic cylinder of the form\({{\bf{y}}^{\bf{2}}}{\bf{ + 2}}{{\bf{x}}^{\bf{2}}}{\bf{ = C}}\) for some constant C. [Hint: Consider the expression for dy/dx implied by Euler’s equations.]

(d) Using the results of parts (b) and (c), argue that the trajectories of this system are closed curves. What does this say about the corresponding solutions?

(e) Figure 5.19 displays some typical trajectories for this system. Discuss the stability of the three critical points indicated on the positive axes.

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.

y''(t)=cos(t-y)+y2(t);y(0)=1,y'(0)=0

In Problems 19 – 21, solve the given initial value problem.

dxdt=2x+y-e2t;    x0=1,dydt=x+2y;   y0=-1

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 – ton air conditioning unit that removes 24,000 Btu/hr. The heat capacity of zone B is12F∘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°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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