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Chapter 5: Q4E (page 249)

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.
x'=x-y,y'=y-4x

Short Answer

Expert verified

The solutions for the given linear system arext=c1e-t+c2e3t and yt=2c1e-t-2c2e3t.

Step by step solution

01

General form

Elimination Procedure for 2 × 2 Systems:

To find a general solution for the system

L1x+L2y=f1,L3x+L4y=f2,

Where L1,L2,L3and L4are polynomials in D=ddt

a. Make sure that the system is written in operator form.

b. Eliminate one of the variables, say, y, and solve the resulting equation for x(t). If the system is degenerating, stop! A separate analysis is required to determine whether or not there are solutions.

c. (Shortcut) If possible, use the system to derive an equation that involves y(t) but not its derivatives. [Otherwise, go to step (d).] Substitute the found expression for x(t) into this equation to get a formula for y(t). The expressions for x(t), and y(t) give the desired general solution.

d. Eliminate x from the system and solve for y(t). [Solving for y(t) gives more constants----in fact, twice as many as needed.]

e. Remove the extra constants by substituting the expressions for x(t) and y(t) into one or both of the equations in the system. Write the expressions for x(t) and y(t) in terms of the remaining constants.

02

Evaluate the given equation

Given that,

x'=x-y                    …1y'=y-4x                 …2

Let us rewrite this system of operators in operator form:

D-1x+y=0                        …34x+D-1y=0                        …4

Multiply (D-1) on equation (3) and subtract with equation (4). one gets,

D-12x-4x=0D2-2D+1x-4x=0D2-2D-3x=0

Since the corresponding auxiliary equation is r2-2r-3=0. The roots arer=-1 and r=3.

Then, the general solution isxt=c1e-t+c2e3t                …5

03

Substitution method

Substitute the equation (5) in equation (3).

D-1c1e-t+c2e3ty=0y=1-Dc1e-t+c2e3ty=c1e-t+c2e3t-ddtc1e-t+c2e3t=c1e-t+c2e3t--c1e-t+3c2e3t=2c1e-t-2c2e3t

Thus, the solutions for the given linear system arext=c1e-t+c2e3t and yt=2c1e-t-2c2e3t.

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Most popular questions from this chapter

In Problems 11–14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).

dxdt=3ydydt=2x

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.

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

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⩾0.

A Problem of Current Interest. The motion of an ironbar attracted by the magnetic field produced by a parallel current wire and restrained by springs (see Figure 5.17) is governed by the equation\(\frac{{{{\bf{d}}^{\bf{2}}}{\bf{x}}}}{{{\bf{d}}{{\bf{t}}^{\bf{2}}}}}{\bf{ = - x + }}\frac{{\bf{1}}}{{{\bf{\lambda - x}}}}\) for \({\bf{ - }}{{\bf{x}}_{\bf{o}}}{\bf{ < x < \lambda }}\)where the constants \({{\bf{x}}_{\bf{o}}}\) and \({\bf{\lambda }}\) are, respectively, the distances from the bar to the wall and to the wire when thebar is at equilibrium (rest) with the current off.

  1. Setting\({\bf{v = }}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}\), convert the second-order equation to an equivalent first-order system.
  2. Solve the related phase plane differential equation for the system in part (a) and thereby show that its solutions are given by\({\bf{v = \pm }}\sqrt {{\bf{C - }}{{\bf{x}}^{\bf{2}}}{\bf{ - 2ln(\lambda - x)}}} \), where C is a constant.
  3. Show that if \({\bf{\lambda < 2}}\) there are no critical points in the xy-phase plane, whereas if \({\bf{\lambda > 2}}\) there are two critical points. For the latter case, determine these critical points.
  4. Physically, the case \({\bf{\lambda < 2}}\)corresponds to a current so high that the magnetic attraction completely overpowers the spring. To gain insight into this, use software to plot the phase plane diagrams for the system when \({\bf{\lambda = 1}}\) and when\({\bf{\lambda = 3}}\).
  5. From your phase plane diagrams in part (d), describe the possible motions of the bar when \({\bf{\lambda = 1}}\) and when\({\bf{\lambda = 3}}\), under various initial conditions.

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=αV-V2t whereα 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=αV-V2t-KV1t,dV2dt=KV1t-R3

b. Find a general solution for the system in part (a) whenα=5min-1,V=20L,K=2min-1, and R3=10  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→+∞?
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