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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鈥�鈥�or鈥�鈥�鈥�(-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)鈫�鈭�鈥�鈥�as鈥�鈥�t鈫�鈭�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鈫�+鈭�.

dxdt=-x+位测,dydt=x-y

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

x'+y'-x=5,x'+y'+y=1

Use the Runge鈥揔utta algorithm for systems with h= 0.1 to approximate the solution to the initial value problem.

x'=yz;x(0)=0,y'=-xz;y(0)=1,z'=-xy2;z(0)=1,

At t=1.

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

x''+y''-x'=2t,x''+y'-x+y=-1

In Problems 1鈥�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
See all solutions

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