tb) Choose a nonzero eigenvector v1. Show that (A -).1)2v :0 for every vector v that is a multiple of v1.

lc) Suppose that v is not a multiple of vr. Show that vand v1 form a basis ofR2.

rdl Set w : (A -.11)v. Show that there are numbers aand b such that w : avt * bv.

re) Show that (A – 1.1)w : Dw, and conclude from thefact that .1. is the only eigenvalue that b : 0.

r0 Conclude that (A – i./)2v:0.Figure 7 shows two tanks, each containing 500 gallons ofa salt solution. Pure water pours into the top tank at a rateof 5 gal/s. Salt solution pours out of the bottom of thetank and into the tank below at a rate of 5 galls. There isa drain at the bottom of the second tank, out of which salt>olution flows at a rate of 5 galls. As a result, the'amountof solution in each tank remains constant at 500 gallons.Initially (time r : 0) there is 100 pounds of salt present inthe first tank, and zero pounds of salt present in the tankimmediately below.

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5 galls

Figure 7. Tlvo cascaded tanks.ta) Set up, in matrix-vector form, an initial value problem

that models the salt content in each tank over time.tb) Find the eigenvalues and eigenvectors of the coeffi-

cient matrix in part (a), then flnd the general solutionin vector form. Find the solution that satisfies the ini-tial conditions posed in part (a).

t c) Plot each component of your solution in part (b) over aperiod of four time constants (see Section 4.7 or Sec-tion 2.2, Exercise 29) 10,47,.). What is the eventualsalt content in each tank? W!y!9iv,e b_qth a physical

and a mathematicalreason-Tor your answer.{Jfigure 8 shows two tanks, each containing 360 liters of

9.2 Planar Systems 391

5 Limin. There are two pipes connecting tank A to tankB. The first pumps salt solution from tank B into tank Aat a rate of 4 L/min. The second pumps salt solution fromtank A into tank B at a rate of 9 L/min. Finally, there isa drain on tank B from which salt solution drains at a rateof 5 L/min. Thus, each tank maintains a constant. volumeof 360 liters of salt solution. Initially, there are 60 kg ofsalt present in tank A, but tank B contains pure water.

.

C-+ 5 L/min !.1 . a .1. ;l

Fisure 8. Two interconnected tanks.(a) Set up, in matrix-vector form, an initial value problem

that models the salt content in each tank over time.

(b) Find the eigenvalues and eigenvectors of the coeffl-cient matrix in part (a), then find the general solutionin vector form. Find the solution that satisfles the ini-tial conditions posed in part (a).

(c) Plot each component of your solution in part (b) over aperiod of four time constants (see Section 4.7 or Sec-tion 2.2, Exercise 29) [0,47,.). What is the eventualsalt content in each tank? Why? Give both a physicaland a mathematicakeason for your answer.

60. In Exercisr u were given the circuit inFigure 9 anTE 6-w that the voltage V across thecapacitor and the current 1 across the inductor satisfled thesystem

V,- V *1.RCC

Suppose that the resistance is R : 112 ohm, the capaci-tance is C : 1 farad, and the inductance is L : I l2henry.If the initial voltage across the capacitor is V(0) : 10volts and there is no initial curent across the inductor,solve the system to determine the voltage and current as afunction of time. Plot the voltage and current as a functionof time. Assume current flows in the directions indicateil.

Figure 9. A parallel circuit withcapacitor, resistor, and inductor.

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rost imPo:ral is quitngle eigetd. We ne

).1)2v 1- r salt solution. Pure water pours into tank A at a rate of

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5 L/min

),ue prothe girc