Question

Consider two brine tanks connected as shown in figure I below. Tank 1 contains xit) pounds of salt in 100 wal of brine and tank 2 contains vit) pounds of salt in 200 gal of brine. The brine in each tank is kept uniform by stirring, and brine is pumped from each tank to the other at the rates indicated in figure 1. In addition, fresh water flows into tank 1 at 20 gal/min, and the brine in tank 2 flows out at 20 gal/min (so the total volume of brine in each of the two tanks remains constant over time 20 gal/min Fresh water 10 gal/min (1) lb y(1) lb 200 gal 100 gal 30 gal/min Tunk 1 Tank 2 1 20 gal/min Figure 1: Two Brine Tanks 1. (10 points) Write down a system of first order linear differential equations that models the amount of salt X(t) and y(t) in the two tanks. Do not solve it. 2. (6 points) Find the associated matrix and calculate its eigenvalues. Do not use a calculator or computer. You do not need to approximate radicals. 3. (4 points) Looking at the eigenvalues, what can you conclude about the long term behavior of x(t) and y(t)?

Answer 1

1. For tank 1 the mass balance of the amount of salt is given as 经 - 10 [ ]- ] - 30[/ min 能 器 「馬」 (1) Likewise, for tank 2 单 - 30 [副]- ]- [] [圖] - 20[黑]是[ ] 三益是「影」 3 4 : 卷 - - 一品) (2)

(2) The associated matrix is given as T - 3/10 1/201 A = -3/20 3/10 57 19 The eigenvalues associated with the matrix A are given by = 0 (1+3)(1+z) - Žeo = 0, The roots are Aq=- o - sve and 12 = - 2 + y2 Note : 12+ + žo ) * + 200 - 50 12 2 + 9 * 70

10) Since the eigen values are both real and negative therefore the solution (5, ) = (0,0) is asymp- totically stable such an equilibrium point is node. 50, the long term behavior is that the solution curves tends to the origin along the eigen vector corresponding to eigen value 12. Hence the amount of salt in both tanks tends to zero as time tends to infinity.