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Unformatted text preview: Math 17C
Kouba
Discussion Sheet 6 1.) Consider the two tanks containing salt water solutions and connected as shown in the
diagram. Tank 1 holds 30 gallons of salt water solution. Tank 2 holds 45 gallons of salt
water solution. Let m1 and 2:2 represent the pounds of salt in Tank 1 and Tank 2, resp.,
at time t. Initially, Tank 1 contains 10 pounds of salt and Tank 2 contains 25 pounds of
salt. The mixture in each tank is kept uniform by stirring, and the mixtures are pumped
from each tank to the other at the rates indicated in the ﬁgure. In addition, fresh water is
pumped‘into Tank 1 at the rate of 15 gal. / min.; the mixture leaves Tank 2 at 15 gal. / min.
Set up and solve a system of differential equations with initial conditions, which represents
the amount of salt in each tank. How much salt is in each tank t = 10 minutes ? glue x‘:u,¢.rgmm ‘rwk 1 MM?
xazun, c‘MMTMZ «*M‘t‘
‘E: Fv‘e,sla warten O [S'JqL/MIVL 2.) Consider the two tanks containing Snapple tea and honey mixtures and connected as
shown in the diagram. Tank 1 holds 200 gallons of mixture. Tank 2 holds 300 gallons of
mixture. Let m1 and 3:2 represent the pounds of honey in Tank 1 and Tank 2, resp., at
time t. Initially, Tank 1 contains 60 pounds of honey and Tank 2 contains 100 pounds
of honey. The mixture in each tank is kept uniform by stirring, and the mixtures are
pumped from each tank‘to the otherlat the rates indicated in the ﬁgure. In addition, a
mixture containing 1/2 pound of honey per gallon is pumped into Tank 1 at 3 gal. / min.; a
mixture containing 3/4 pound of honey per gallon is pumped into Tank 2 at 2 gal. / min.;
the mixture leaves Tank 2 at 7 gal. /min. SET UP, BUT DO NOT SOLVE, a sytem of
differential equations with initial conditions, which represents the amount of honey in each tank. PAY CLOSE ATTENTION TO FLOW RATES IN AND OUT OF EACH TANK !! 3.) Assume that a 21—year old college student ingests four consecutive shots of liquor
containing a total of 2 ounces of ethanol (grain alcohol). After 2 hours there is 1 /2 ounce
of ethanol in the person’s body tissue. Let .131 and 302 be the ounces of ethanol in the
person’s body tissue and urinary tract, resp., at time 15 hours. Use the mathematical
model from class to determine formulas for :31 and :32. How much ethanol has passed into
the urinary tract after t = 5 hours ? 4.) A single dose of quinine sulfate (an antimalarial drug) is administered to a patient.
A laboratory measurement shows that the urinary tract has accumulated 171 mg. of the drug after 5 hours. Let $1 and 11:2 be the mg. of quinine sulfate in the person’s body d3:
tissue and urinary tract, resp., at time t hours and assume that —d—tl : —.15m1. Use the mathematical model from class to determine formulas for 331 and $2. Determine the initial
dosage in mg. of quinine sulphate. 5.) Find the Jacobi Matrix for each function. a.) ﬂay): b.) f($,y):( “63’ > 1n(."c  y)
c) f(:v,y) = d.) f(w,y)=<co:zr51§cxg)y)) e.) f(:r,y)= Determine the linearization, L(.r), for : m2(.r — 1) at :1: = 2. Use L(:r) to estimate
the value of f at as = 1.9. 7.) Determine the linearization, L(x,y), for f($,y) = 3:2 + 2312 at (x,y) = (1,—1). Use
L(:c, y) to estimate the value of f at (2:,y) = (1.1, —0.9). 2:13 + 63;
m2 + 2y
of the points (1, 1), (—2/3, —1), and (—1, 1). Compute the eigenvalues for each matrix. 8.) Consider the function f (3:, y) = < > . Determine the Jacobi Matrix for each 9.) Consider the 2x2 linear systems of differential equations given by X ’ = AX, each of
which has (0,0) as an equilibrium . For each matrix A below, determine if (0,0) is stable or unstable. Then categorize (0,0) as a sink, source, saddle, stable spiral, unstable spiral,
or neutral spiral. a.)A=<,11 b)A=(_02 33) C)A:<11/2 we 2) a M 2:) 19> 10.) VVithout explicitly computing the eigenvalues of A, determine whether the real parts
of both eigenvalues are negative. a.)A=(:g b)A=<_02 0)A=(‘11 33) 11.) Determine all equilibria for each of the following 2x2 nonlinear systems of differential
equations. dill a.) gzml—xz b.) ?=8x1—$%—w1$2
—§t3=x1(a:2—3) §E=4$2—$1$2+$§
dwl c) it— —— ﬁxg + x3
dfltz —— =$1$2 ——:r§ +2zc1 dt 12.) Find all zero isoclines for the systems in problem 11. Plot and label these isoclines d d
using % = 0 or ~32 2 0 in the wag—coordinate system. Also label the equilibria. dt 13.) Set up a table and direction ﬁeld for the following nonlinear system of differential
equations using slopes, direction vectors, and speed for the following points (1121,1112) : (0:0)(0,1),(02),(1,0),(1,1)a(1a2)a(2a0)a(2,1),(2,2) dt _ 2
dCBQ ‘23:]: 14.) Find all equilibria for each system of differential equations, and use the analytical
approach to determine the stability of and classify each equilibrium. . — = — — b. — = '
a) ddt $1 331 $1JJ2 ) ddt mm + 11:2
x a:
E223mg—11211I2—21‘g ﬁz(L‘l—m'g 15.) Determine the equilibria for the following nonlinear system. Graph the zero isoclines.
Use the graphical approach to determine the stability of each equilibrium, if possible. —— 2 CL' — (1)
dt 1 2
dCL'g 433'ch = f1($1,$2) i=2; _
dt — f2($1,332)
of the following zero isocline diagrams, assume that both f1 > O and f2 > 0 near the 16.) Consider the system of differential equations given by { . For each 4 origin. Complete the twodimensional sign chart for f1 and f2 using the isocline hopping
method (i.e., by assuming the functions change sign when they cross their own isoclines).
Then set up a signed Jacobi Matrix to determine (if possible) the stability of the given equilibrium, where the isoclines cross. 17.) Show that (5,3) is an equilibrium for the following nonlinear system of differen
tial equations. Complete the twodimensional sign chart for f1 and fg‘ using the isocline
hopping method (i.e., by assuming the functions change sign when they cross their own
isocline). Then set up a signed Jacobi Matrix to determine (if possible) the stability of the equilibrium, (5, 3). drc
C—ld—tl =  132)
(If
(1—: = (132(121 ~CE2  +++++++++++++++++++++++++++++++++ ”I almost had a psychic girlfriend, but she left me before we met.” — Steven Wright +++++++++++++++++++++++++++++++++ ” Nothing will work unless you do.” — Maya Angelou ...
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 Summer '09
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