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Chapter 9: Q46E (page 442)

Use theorem 9.3.13 to solve the initial value problem

dx→dt=231012001x→with x→(0)=[21-1].

Hint: Find first x3(t), then x2(t), and finally x1(t).

Short Answer

Expert verified

The solution isen2-3t+t2-1+2tt

Step by step solution

01

Definition of first order linear differential equation.

Consider the differential equation f'(t)-af(t)=g(t)whereg(t)is a smooth function and 'a'is a constant. Then the general solution will bef(t)=eat∫e-atg(t)dt.

02

Determination of the solution.

Consider the differential equation as follows.

dx→dt=231012001x→A=231012001A=2f't-2ft=0

Now, the differential equation is in the form as follows.

f'(t)-af(t)=g(t), where g(t) is a smooth function, then the general solution will be as follows.

f(t)=eat∫e-atg(t)dt

03

Compute the calculation of the solution.

Substitute the value 0 for g(t) and 2 for a inf(t)=eat∫e-atg(t)dtas follows.

role="math" localid="1660805430798" f(t)=eat∫e-atg(t)dtf(t)=e2t∫e-2t×0×dtf(t)=e2t∫0×dtf(t)=e2t·C

Here C is constant

Also it can be written as follows.

C'=2-3t+t2-1+2tt

The solution ise2t2-3k+t2-1+2tt

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Most popular questions from this chapter

The displacement x(t) of a certain oscillator can be modeled by the DE

d2xdt2+6dxdt+9x=0.

Find the solution x(t) for the initial value x(0) = 0,x' (0) = 1. Sketch the graph of the solution. How many times will the oscillation go through the equilibrium state x = 0in this case?

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feedback Loops:Suppose some quantitieskix1(t),x2(t),...,xn(t)can be modelled by differential equations of the form localid="1662090443855">|\begingathereddx1dt=-k1x                 -bxn\hfilldx2dt=x1-k2                      \hfill.\hfill.\hfilldxndt =                           xn-1-knxn\hfill\endgathered|

Where b is positive and the localid="1662090454144">ki are positive.(The matrix of this system has negative numbers on the diagonal, localid="1662090460105" 1's directly below the diagonal and a negative number in the top right corner)We say that the quantities localid="1662090470062">x1,x2,...,xndescribe a (linear) negative feedback loop

  1. Describe the significance of the entries in this system inpractical terms.
  2. Is a negative feedback loop with two componentslocalid="1662090478095">(n=2) necessarily stable?
  3. Is a negative feedback loop with three components necessarily stable?
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