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Chapter 5: Q27E (page 272)

Using the software, sketch the direction field in the phase-plane for the system dxdt=-2x+y,dydt=-5x-4y. From the sketch, predict the asymptotic limit (ast→∞

of the solution starting at (1, 1).

Short Answer

Expert verified

The solution starts at (1,1) and moves towards zero as t→∞ the solution approaches to (0,0).

Step by step solution

01

Find the critical point

Here the equation is:

dxdt=-2x+ydydt=-5x-4y

And

dydx=-5x-4y-2x+y

The solution starts at (1,1) and moves towards zero as the solution approaches to (0,0).

02

Sketch the Directional field

This is the required result.

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Most popular questions from this chapter

In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

D+1u-D+1v=et,D-1u+2D+1v=5

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

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

In Problems 25 – 28, use the elimination method to find a general solution for the given system of three equations in the three unknown functions x(t), y(t), and z(t).

x'=x+2y+z,y'=6x-y,z'=-x-2y-z

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).

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)
See all solutions

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