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Chapter 5: Q15RP (page 306)

Sketch some typical trajectories for the given system and by comparing with Figure\({\bf{5}}{\bf{.12}}\), page\({\bf{267}}\), and identify the type of critical point at the origin.

\(\begin{array}{c}{\bf{x}}'{\bf{ = - 2x - y}}\\{\bf{y}}'{\bf{ = 3x - y}}\end{array}\)

Short Answer

Expert verified

The critical point \(\left( {{\bf{0,0}}} \right)\) is an asymptotically stable spiral point.

Step by step solution

01

Finding the value of \({\bf{x,y}}\)

One can solve the critical point. To do so we need to solve the system\({\bf{x}}'{\bf{ = 0,y}}'{\bf{ = 0}}\), so one has

\(\begin{array}{c}{\bf{0 = - 2x - y}}\\{\bf{0 = 3x - y}}\end{array}\)

The first equation gives us that\({\bf{y = - 2 x}}\), so substituting this into the second equation one gets that\({\bf{5 x = 0}}\).

02

Finding the critical point

So, one has that \({\bf{x = 0, y = 0}}\) and the critical point is\(\left( {{\bf{x, y}}} \right){\bf{ = }}\left( {{\bf{0,0}}} \right)\).

Comparing this picture with the Figure \({\bf{5}}{\bf{.12}}\) in the Textbook one can conclude that the critical point \(\left( {{\bf{0,0}}} \right)\) is an asymptotically stable spiral point.

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Most popular questions from this chapter

Two large tanks, each holding 100 L of liquid, are interconnected by pipes, with the liquid flowing from tank A into tank B at a rate of 3 L/min and from B into A at a rate of 1 L/min (see Figure 5.2). The liquid inside each tank is kept well stirred. A brine solution with a concentration of 0.2 kg/L of salt flows into tank A at a rate of 6 L/min. The (diluted) solution flows out of the system from tank A at 4 L/min and from tank B at 2 L/min. If, initially, tank A contains pure water and tank B contains 20 kg of salt, determine the mass of salt in each tank at a time t猢�0.

Feedback System with Pooling Delay. Many physical and biological systems involve time delays. A pure time delay has its output the same as its input but shifted in time. A more common type of delay is pooling delay. An example of such a feedback system is shown in Figure 5.3 on page 251. Here the level of fluid in tank B determines the rate at which fluid enters tank A. Suppose this rate is given byR1t=伪V-V2t where伪 and V are positive constants andV2t is the volume of fluid in tank B at time t.

  1. If the outflow rate from tank B is constant and the flow rate from tank A into B isR2t=KV1t where K is a positive constant andV1t is the volume of fluid in tank A at time t, then show that this feedback system is governed by the system

dV1dt=伪V-V2t-KV1t,dV2dt=KV1t-R3

b. Find a general solution for the system in part (a) when伪=5min-1,V=20L,K=2min-1, and R3=10鈥�鈥�L/min.

c. Using the general solution obtained in part (b), what can be said about the volume of fluid in each of the tanks as t鈫�+鈭�?

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.

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;鈥�鈥�鈥�鈥�x0=1,dydt=4x-3y+b;鈥�鈥�鈥�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鈫�+鈭� ), or a runaway arms race (x and y approach+鈭� as t鈫�+鈭�).

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
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