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Chapter 5: Q16E (page 271)

In Problems 15–18, find all critical points for the given system. Then use a software package to sketch the direction field in the phase plane and from this description the stability of the critical points (i.e., compare with Figure 5.12).

dxdt=-5x+2y,dydt=x-4y

Short Answer

Expert verified

This is a stable node point is (0,0).

Step by step solution

01

Find critical points

Here the system is;

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

For critical points equate the system equal to zero.

localid="1663967065664" -5x+2y=0x-4y=0

Solve for x and y by eliminating the method.

The values of x=0 and y=0.

So, this is the stable node point (0,0).

02

Sketch

This is the required result.

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Most popular questions from this chapter

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).

d2xdt2-y=0

A building consists of two zones A and B (see Figure 5.5). Only zone A is heated by a furnace, which generates 80,000 Btu/hr. The heat capacity of zone A is per thousand Btu. The time constant for heat transfer between zone A and the outside is 4 hr, between the unheated zone B and the outside is 5 hr, and between the two zones is 2 hr. If the outside temperature stays at , how cold does it eventually get in the unheated zone B?

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), z(t).

x'=3x+y-z,y'=x+2y-z,z'=3x+3y-z

Use the result of Problem 31 to prove that all solutions to the equation\({\bf{y'' + }}{{\bf{y}}^{\bf{3}}}{\bf{ = 0}}\)remain bounded. [Hint: Argue that \(\frac{{{{\bf{y}}^{\bf{4}}}}}{{\bf{4}}}\) is bounded above by the constant appearing in Problem 31.]

Arms Race. A simplified mathematical model for an arms race between two countries whose expenditures for defense are expressed by the variables x(t) and y(t) is given by the linear system

dxdt=2y-x+a;    x0=1,dydt=4x-3y+b;   y0=4,

Where a and b are constants that measure the trust (or distrust) each country has for the other. Determine whether there is going to be disarmament (x and y approach 0 as t increases), a stabilized arms race (x and y approach a constant ast→+∞ ), or a runaway arms race (x and y approach+∞ as t→+∞).
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