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Chapter 5: Q5E (page 259)

In Problems 1鈥�7, convert the given initial value problem into an initial value problem for a system in normal form.

x''+y-x'=2t;x(3)=5,x'(3)=2,y''-x+y=-1;y(3)=1,y'(3)=-1

[hint]x1=x,x2=x',x3=y,x4=y'

Short Answer

Expert verified

x'1(t)=x2(t)x'2(t)=x2(t)-x3(t)+2tx'3(t)=x4(t)x'4(t)=x1(t)-x3(t)-1x1(3)=5,x2(3)=2,x3(3)=1,x4(3)=-1

Step by step solution

01

Express the equation in form of x

Here givenx''+y-x'=2t and y''-x+y=-1.

Rewrite the equationsx''=x'-y+2t andy''=x-y-1

Denote,

x1(t)=x(t)x2(t)=x'(t)x3(t)=y(t)x4(t)=y'(t)

The equation transforms as;

x'1(t)=x2(t)x'2(t)=x2(t)-x3(t)+2tx'3(t)=x4(t)x'4(t)=x1(t)-x3(t)-1

02

the initial conditions

The given initial conditions are x(3)=5,x'(3)=2and y(3)=1,y'(3)=-1.

Initial conditions after transformations x1(3)=5,x2(3)=2,x3(3)=1,x4(3)=-1

This is the required result.

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Most popular questions from this chapter

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.]

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=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�?

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.

D-3x+D-1y=t,D+1x+D+4y=1

Generalized Blasius Equation. H. Blasius, in his study of the laminar flow of a fluid, encountered an equation of the form y'''+yy''=(y')2-1. Use the Runge鈥揔utta algorithm for systems with h = 0.1 to approximate the solution that satisfies the initial conditions y(0)=0,y'(0)=0,y''(0)=1.32824. Sketch this solution on the interval [0, 2].
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