plz solve the problems

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 v and v1 form a basis ofR2.

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

re) Show that (A - 1.1)w : Dw, and conclude from the fact 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 of a salt solution. Pure water pours into the top tank at a rate of 5 gal/s. Salt solution pours out of the bottom of the tank and into the tank below at a rate of 5 galls. There is a drain at the bottom of the second tank, out of which salt >olution flows at a rate of 5 galls. As a result, the'amount of solution in each tank remains constant at 500 gallons. Initially (time r : 0) there is 100 pounds of salt present in the first tank, and zero pounds of salt present in the tank immediately 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 solution in 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 a period of four time constants (see Section 4.7 or Sec- tion 2.2, Exercise 29) 10,47,.). What is the eventual salt 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 tank B. The first pumps salt solution from tank B into tank A at a rate of 4 L/min. The second pumps salt solution from tank A into tank B at a rate of 9 L/min. Finally, there is a drain on tank B from which salt solution drains at a rate of 5 L/min. Thus, each tank maintains a constant. volume of 360 liters of salt solution. Initially, there are 60 kg of salt 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 solution in 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 a period of four time constants (see Section 4.7 or Sec- tion 2.2, Exercise 29) [0,47,.). What is the eventual salt content in each tank? Why? Give both a physical and a mathematicakeason for your answer.

60. In Exercisr u were given the circuit in Figure 9 anTE 6-w that the voltage V across the capacitor and the current 1 across the inductor satisfled the system

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) : 10 volts and there is no initial curent across the inductor, solve the system to determine the voltage and current as a function of time. Plot the voltage and current as a function of time. Assume current flows in the directions indicateil.

Figure 9. A parallel circuit with capacitor, resistor, and inductor.

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

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5 L/min

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