Math 33B Worksheet 8 (Higher Dimensional Systems)

Name: Score:Circle the name of your TA: Ziheng Nicholas VictoriaCircle the day of your discussion: Tuesday Thursday

The last two problems on this worksheet are based o↵ of examples from the following link:

http://www.math.utah.edu/~gustafso/2250systems-de.pdf

1. Solve X0 =

0

@1 �2 2�2 1 �22 �2 1

1

AX.

KEY

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2. Solve X0 =

0

@2 1 60 2 50 0 2

1

AX. Hint: The general solution is of the form

X = c1v1e�t + c2e

�t(v1t+ v2) + c3e�t

✓v1

t2

2+ v2t+ v3

◆

where (A� �I)v1 = 0, (A� �I)v2 = v1, (A� �I)v3 = v2.

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3. Let A,B,C be brine tanks with volumes 60 L, 30 L, 60 L, respectively. Suppose that fluiddrains from tank A to be B at a rate r, drains from tank B to C at rate r, then drains fromtank C to A at rate r. The tank volumes remain constant due to constant recycling of thefluid. Take r = 10 L/min.

Let x1(t), x2(t), x3(t) denote the amount of salt at time t in each tank. No salt is lost fromthe system, due to recycling. The recycled cascade is modeled by the system

x01 = �1

6x1 +

1

6x3

x02 =

1

6x1 �

1

3x2

x03 =

1

3x2 �

1

6x3.

Find the general solution to this system.

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Fideimvalues

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4. Consider a forest having one or two varieties of trees. We select some of the oldest trees, thoseexpected to die o↵ in the next few years, then follow the cycle of living into dead trees. Thedead trees eventually decay and fall from seasonal and biological events. Finally, the fallentrees become humus. Let variables x, y, z, t be defined by

x(t) = biomass decayed into humus,

y(t) = biomass of dead trees,

z(t) = biomass of living trees,

t = time in decades

A typical biological model is

x0(t) = �x(t) + 3y(t)

y0(t) = �3y(t) + 5z(t)

z0(t) = �5z(t)

Suppose there are no dead trees and no humus at t = 0 with initially 80 units of living treebiomass, i.e. x(0) = y(0) = z(0) = 80. Find the solution to this system.