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Chapter 5: Q28E (page 272)

Figure 5.16 displays some trajectories for the system dxdt=y,dydt=-x+x2What types of critical points (compare Figure 5.12 on page 267) occur at (0, 0) and (1, 0)?

Short Answer

Expert verified

The points are (0,0) and saddle point(1,0).

Step by step solution

01

Find the critical point.

Here the equation is:

dxdt=ydydt=-x+x2

And

dydx=-x+x2y

Here the points are (0,0) and saddle point (1,0).

02

Sketch Directional field.

This is the required result.

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Most popular questions from this chapter

The doubling modulo \({\bf{1}}\) map defined by the equation \(\left( {\bf{9}} \right)\)exhibits some fascinating behavior. Compute the sequence obtained when

  1. \({{\bf{x}}_0}{\bf{ = k / 7}}\)for\({\bf{k = 1,2, \ldots ,6}}\).
  2. \({{\bf{x}}_0}{\bf{ = k / 15}}\)for\({\bf{k = 1,2, \ldots ,14}}\).
  3. \({{\bf{x}}_{\bf{0}}}{\bf{ = k/}}{{\bf{2}}^{\bf{j}}}\), where \({\bf{j}}\)is a positive integer and \({\bf{k = 1,2, \ldots ,}}{{\bf{2}}^{\bf{j}}}{\bf{ - 1}}{\bf{.}}\)

Numbers of the form \({\bf{k/}}{{\bf{2}}^{\bf{j}}}\) are called dyadic numbers and are dense in \(\left( {{\bf{0,1}}} \right){\bf{.}}\)That is, there is a dyadic number arbitrarily close to any real number (rational or irrational).

In Problems 7鈥�9, solve the related phase plane differential equation (2), page 263, for the given system.

dxdt=x2-2y-3,dydt=3x2-2xy

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

dxdt=-8y,dydt=18x

In Problems 23 and 24, show that the given linear system is degenerate. In attempting to solve the system, determine whether it has no solutions or infinitely many solutions.

D-1x+D-1y=-3e-2t,D+2x+D+2y=3et

Find all the critical points of the system

dxdt=x2-1dydt=xy

and the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?
See all solutions

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