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Chapter 9: Q26E (page 440)

For the linear system dx鈫�dt=[-2031]x鈫�(t)find the matching phase portrait.

Short Answer

Expert verified

The phase portrait corresponds toV

Step by step solution

01

To find the eigenvalues

Consider the linear system as follows.

dx鈫�dt=-2031x鈫�t

To find the eigenvalues, evaluate A-位I=0as follows.

A-位I=0-2-位031-位=0-2-位1-位-30=0位2-2+2位-位-0=0

Simplify further as follows.

位2+位-2=0位2-位+2位-2=0位位-1+2位-1=0位+2位-1=0

Simplify further as follows.

位+2=0位1=-2位-1=0位2=1

Therefore, the eigenvalues are 位1=-2and 位2=1

02

Observation of the phase portrait

The eigenvalues are 位1=-2and位2=1

Since, the eigenvalues are real and distinct are of opposite sign.

Therefore, the phase portrait is in figure 5 as follows.

Hence, the phase portrait to the linear system dx鈫�dt=-2031x鈫�tis V.

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Most popular questions from this chapter

Consider a2脳2matrixAwith eigenvalues卤蟺i. Let v鈫�+iw鈫�be an eigenvector of A with eigenvalue . Solve the initial value problem dx鈫�dt=Ax鈫�withx鈫�0=w鈫�.

Draw the solution in the accompanying figure. Mark the vectorsx鈫�(0),x鈫�(12),x鈫�(1),andx鈫�(2).

Solve the differential equationf"(t)=0and find all the real solutions of the differential equation.

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation dxdt=fxas dxfx=dtand integrate both sides.

9.dxdt=xk(with鈥�k鈮�1),x(0)=1

Question: Consider the interaction of two species in a habitat. We are told that the change of the populationsx鈫�(t)andy鈫�(t)can be moderated by the equation

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

,

where timeis a measured in years.

  1. What kind of intersection do we observe (symbiosis, competition, or predator-prey)?
  2. Sketch the phase portrait for the system. From the nature of the problem, we are interested only in the first quadrant.
  3. What will happen in the long term? Does the outcome depend on the initial populations? If so, how?

Solve the differential equationf''(t)+f'(t)-12f(t)=0and find solution of the differential equation.
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