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Chapter 5: Q23E (page 271)

In Problems 19–24, convert the given second-order equation into a first-order system by setting v=y’. Then find all the critical points in the yv-plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12).

y''(t)+y(t)-y(t)4=0

Short Answer

Expert verified

The critical points are (0,0),(1,0).

Step by step solution

01

Find the critical point

Here the equation is y''t+yt-yt4=0.

Put v=y'  and  v'=y''.

Then the system is;

y'=vy''=-y+y4v'=-y+y4

For critical points equate the system equal to zero.

v=0-y+y4=0y-1+y3=0y=0 or -1+y3=0

If y≠0  then

-1+y3=0y=1

So, the critical point is (0, 0) and (1, 0).

02

Sketch the directional field. 

Therefore, the critical points are (0, 0), (1, 0).

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Most popular questions from this chapter

In Problems 3–6, find the critical point set for the given system.

dxdt=y2-3y+2,dydt=(x-1)(y-2)

Show that the Poincare map for equation (1) is not chaoticby showing that if\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)are two initial values that define the Poincare maps\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\) and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\), respectively, using the recursive formulas in (3), then one can make the distance between\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\)small by making the distance between\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\) and \({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)small. (Hint: Let \({\bf{(A,}}\phi {\bf{)}}\)and \({\bf{(}}{{\bf{A}}^{\bf{*}}}{\bf{,}}{\phi ^ * }{\bf{)}}\) be the polar coordinates of two points in the plane. From the law of cosines, it follows that the distance d between them is given by\({{\bf{d}}^{\bf{2}}}{\bf{ = (A - }}{{\bf{A}}^{\bf{*}}}{{\bf{)}}^{\bf{2}}}{\bf{ + 2A}}{{\bf{A}}^{\bf{*}}}{\bf{(1 - cos(}}\phi {\bf{ - }}{\phi ^ * }{\bf{))}}\).)

In Problem 36, if a small furnace that generates 1000 Btu/hr is placed in zone B, determine the coldest it would eventually get in zone B has a heat capacity of 2°F per thousand Btu.

In Problems 11–14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).

dxdt=3ydydt=2x

Find the critical points and solve the related phase plane differential equation for the system dxdt=(x-1)(y-1),dydt=y(y-1).Describe (without using computer software) the asymptotic behavior of trajectories (as role="math" localid="1664302603010" t→∞) that start at

  1. (3, 2)
  2. (2, 1/2)
  3. ( -2, 1/2)
  4. (3, -2)
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