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Chapter 5: Q3E (page 259)

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

y4(t)-y3(t)+7y(t)=cost;y(0)=y'(0)=1,y''(0)=0,y3(0)=2

Short Answer

Expert verified

x'4(t)=x4-7x1+cost

Step by step solution

01

express the equation in form of x

Here given

y4(t)-y3(t)+7y(t)=cost

Denote,

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

The equation transforms as;

x'1(t)=x2(t)x'2(t)=y''(t)=x3(t)x'3(t)=y'''(t)=x4(t)x'4(t)=y4(t)x'4(t)=x4-7x1+cost

02

the initial conditions

The given initial conditions arey(0)=y'(0)=1,y''(0)=0,y3(0)=2

Initial conditions after transformations;

x1(0)=1x2(0)=1x3(0)=0x4(0)=2

This is the required result.

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Most popular questions from this chapter

Let A, B and C represent three linear differential operators with constant coefficients; for example,

A:=a2D2+a1D+a0,B:=b2D2+b1D+b0,C:=c2D2+c1D+c0,

Where the a鈥檚, b鈥檚, and c鈥檚 are constants. Verify the following properties:

(a) Commutative laws:

A+B=B+A,AB=BA

(b)Associative laws:

A+B+C=A+B+C,ABC=ABC.

(c)Distributive law:AB+C=AB+AC

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

y6(t)=y'(t)3-sin(y(t))+e2t;y(0)=y'(0)=......=y(5)(0)=0

In Problems 3鈥�6, find the critical point set for the given system.

dxdt=x2-2xy,dydt=3xy-y2

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�?

The doubling modulo \({\bf{1}}\) map defined by the equation \(\left( {\bf{9}} \right)\)exhibits some fascinating behavior. Compute the sequence obtained when

  1. \({{\bf{x}}_0}{\bf{ = k / 7}}\)for\({\bf{k = 1,2, \ldots ,6}}\).
  2. \({{\bf{x}}_0}{\bf{ = k / 15}}\)for\({\bf{k = 1,2, \ldots ,14}}\).
  3. \({{\bf{x}}_{\bf{0}}}{\bf{ = k/}}{{\bf{2}}^{\bf{j}}}\), where \({\bf{j}}\)is a positive integer and \({\bf{k = 1,2, \ldots ,}}{{\bf{2}}^{\bf{j}}}{\bf{ - 1}}{\bf{.}}\)

Numbers of the form \({\bf{k/}}{{\bf{2}}^{\bf{j}}}\) are called dyadic numbers and are dense in \(\left( {{\bf{0,1}}} \right){\bf{.}}\)That is, there is a dyadic number arbitrarily close to any real number (rational or irrational).
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