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Unformatted text preview: Section 6.2 6.2 Problems Sketching SecondOrder DES In Problems 1—4, ﬁnd the con~
stant soiution(s) x(r) E kfor each secondorder equation, and
determine the behavior asfoit'ows: (a) Rewrite the equation as a system of two ﬁrstorder
equations. 0)) Find the equilibrium soiution(s) of the equivalent
ﬁnst«order system. (c) Deduce the behavioroj the trajectories about the ﬁxed
point as t > oo—for‘ example, ﬂying away from the
ﬁted point, orbiting it, approaching it. ((1) Describe the physical behavior of solutions to the DE.
Tell what it means . for a massnspring system. 1.x”+.x’+x=0 2. x”—x’—§xm0 .3. x” +.x =1 4. x” +2x’ +x = a Matching Game Match each system of Probiems 5—8 with
one ofthe vector‘ﬁel'ds in Fig. 6.2.14. HiNT: Use infonnation
from eigenvalues (Sec. 5 .3 ) and nuilclines (Sec. 2. 5).. 5.x’mx 6. x'=—.x
y’=y y’mmy i_ 0 1i s,_1 1
7.x._[l_1x 8.x._1__1x Solutions is: General Find the general solutions for Prob~
”tents 9—22. Sketch the eigenvectors and a few typical trajecto—
ries. (Show your method. ) steel“: jlx 10. gel"; élx
11.i=[; ”tile 12. gal“; :13]
13.2:[2 —%]x 143ml; 3:]? . l 16. i'zlj j}: (w
rrrﬂfw‘
rrrr‘“""
,rﬂhmm ‘~ \l
\\
.—.~~~‘"~\ '\ K
"hummxk \\
\\
\\ «ﬁmmxx n_._an~~\\ \ (A) (13)
FIGURE 6.2 Linear Systems with Real Eigenvalues ..,__ 3 W2“
17.x...»[2 _2]x
19 , 142. “x: 3 ”.4" .. 4 ——3 W.
7 r
1. x [8 _6]x 18. 20. 369 22.x Repeated Eigenvalues Find the general solutions for Prob
lems 23 and 24. Sketch the eigenvectons and a few typical
trajectories (Show your method. ) .. —~1 i ..
7 !_
3.x _[ 4 3]); 24. if m l—3 iii Solutions in ?articular Soive the WP: in Probiems 2564. Sketch the trajectory.  .. —2 I ...
? ’—
..3. x l 5 Ax, _, 1 —3 _.
26.x:[w2 2]X,
.. 2 O _ a 1. 7.x.—[0 4x,
,_ m2 4 a 28.x_[ 1 11x,
_. i 1 _ ‘H 1.... ..).1:;~..[E 1]x,
a,_ ~—3 2 m 38.x[ 1 “2 X. 3}. i’—[”2 I]? . . _ 4 m2 .,
",W i. 12 .. 32.x".[3 llx, .1 4 Vector ﬁelds to match to the systems in Problems 5w8. iiﬂ) = 52(0)m Rim: §(O)m KO): 21(0): §(O)m 5K0}: lél {a Irrrr/l
frif/{I
(In/{KI 222/22, 1222/2; .370 Chapter 6 E Linear Systems of Differential Equations .33. a: [i ills: 34a[13]t
“ m 21" .35. Creating New Probiems (:1) Find a 3 x 3 matrix with a double eigenvalue AI = 12
that has only one eigenvector. and a separate eigenvaiue
l3 # M with another eigenvector. (in) Find a 3 x 3 matrix with a triple eigenvalue and two
linearly independent eigenvectorsl 36. Repeated Eigenvalue Theory Suppose that (a) Show that the system has a double eigenvalue if' and
oniy if the condition (a —— (1)3 + 4bc = 0 is satisﬁed,
and that the eigenvalue is i“! + d) (b) Show that if the condition in (a) holcis and a = d, the
eigenspace wiii be altooimensiona] only if the mattix l? 3i (c) Show that if the condition in (:1) holds and a gé d,
the eigenvectors belonging to %(o t— (2’) ate linearly
dependent; that is, scalar multiples of 2!)
t1  o (d) Show that the general solution of the system with doom
ble eigenvalue and a # d is A 2b A 21) 0
Cleiid—a +c2e' {time + 2 ‘ where A = i“? ~t~ d). is diagoaal. 37. Quick Sketch For equation (10) of Example 6, show the
calculations for vectors at whatever points you choose.
Conﬁrm that your tesult has the same chat‘actetistics as
Fig. 6.2 7. Then sketch some trajectories on your graph. 38, Generalized Eigenvectors Suppose that we wish to em
tend the method described for ﬁnding one generalized
eigenvector to ﬁnding two (or more) genetalized eigen
vectors. Let’s look at the case where A has multiplicity
3 but has only one linearly independent eigenvector v.
First, we find {1] by the method descnbed in this section.
Then we ﬁnd ﬁg such that (A —— snag = a], or (A _ Aliza»; = tr (We continue in this fashion to obtain it, €11,113, , , in,
for r < m, where m is the multiplicity of Pt and r is the
number of“‘rnissing" eigenvectors for it.) (a) Show that i1: Gui}, i1 =(f§+ﬁ1)eh, _ l _ _ 
x3 : (gray +111, + L13)?“ ate solutions of i' 2 AR, given that A? m H and it
has multiplicity .3 and r = 2. (b) Show that the vectors i1, i2, and i; ate linearly
independent. 1 i I
(c) Solve 21’: O i l i.
D 0 l 39. One Independent Eigenvector Consider i’ = A)? for O 0 1
Am 1 0 NB.
6 i 3 (a) Show that A has eigenvalue A m i with multiplicity
3, and that all eigenvectors are scalar multiples of at (b) Lise part (a) to ﬁnd a solution of the system in the form 1=€ri Kl (c) Find a second solution in the form
it; = in??? + e’ii,
whet‘e vector 11 is to be determined. HINT: Find it that satisﬁes (A — Ilﬁ = a. (d) Use the result of part (c) to ﬁnd a third solution of the
system of the form  1".” 1 r—
x3: 3t‘6v+teu+etv. HENT: Find a that satisﬁes (A — DVV 2 ii. Solutions in Space Find the general solutions for Problems 40
and 41 . 3 2 2
40.§’m[1 4 1i
—2 «Li —1 —1 i 0
41. i’: l 2 1 i
0 3 —i Section 6.2 Spatial Particulars Obtain solutions for the [VPS of Pro!»
[ems 42 otic143. 42‘ 43. 44. 46. l *1 {l D
i': 0 ——l 3 3}, H0)“: 0
—1 i 0 l t t 0 2
She [I I 0 it, 17(0) 2 [it]
0 0 ml 2 Veriﬁcation of Independence Show that the solutions
obtained in Example 7 of this section ate linearly
independent, . Adjoint Systems The iinearsystem i l 9)
has a “cousin" system W : nATﬁl’ (20) called its adjoin: (Taking the negative of the transpose
twice returns the original matrix, so each system is the
adjoint of the other) (:1) Determine the system adjoint to i’ = [C1, 5] it. (b) Establish that for the solutions 5% and a of adjoint sys
terns (l9) and (20) it is true that d _ “r w —— w x m l i so ﬁfty; a constant HINT: (AB)T m BTAT.
Solve the 1V? consisting of the system of part (a) and ma Soive the IV? consisting of the adjoint of the system
of part (a) and the initial condition etc) a m For the initial conditions in parts (c) and (d),
wT(0}i(O) :2 0. What can you conclude about the
paths 17:0} and Mt} it'they are plotted on the same set
of'axes? ~rT~ ~T~t =w x+w x :0, (C) (d) (e) CauchyEuler Systems The system ti’ = Ai, where A
is a constant matrix and t > O, is called a CauchyEuler
system (a) Show that the Cauchwaulet‘ system has a solution of the form SE n Mir, wheie A is an eigenvalue of A and
h is a corresponding eigenvector. (b) Solve the CauchyEuler system .. , ml 
tx’=[3 W7]x, t>0. Linea: Systems with Real Eigenvalues 371 Computer Lab: Fredicting Phase Portraits Check your in» tuition for each of the .syistetizs in Problems 47—50, following
these steps. (a) Sketch what you think the direction ﬁeld looks like.
(b) Use an open—ended solver to draw the vector ﬁeld (c) Solve the systettt ottnlyticollynnd compare with results
in (a) and (bi Explain or" reconcile any (inferences 47. x’ = x 48. ,t’ = 0
y’ = iv y’ I ~y 49. x’ = x + y 50' .x’ = y
j.” m .x + y y" z x 51. Radioactive Decay Chain The radioactive isotope of lo— (1: Is.) dine, 1—135, decays into the radioactive isotope Xe—EBS
of xenon; this in turn decays into another (stable) prod—
uct. The halflives of iodine and xenon are 67 home anti
9?. hours, respectively (it) Write a system of differential equations describing the
amounts of ll35 and Xe~135 present at any time (b) Obtain the general solution of the system faund in
part (a) . Multiple Compartment Mixing I Consider two large tanks, connected as shown in Fig. 6 2.15. Tank A is ini—
tially ﬁlled with 100 go! oi'water in which 25 lb oi'sait has
been dissolved. Tank B is initially ﬁtted with 100 get of
pure water. Pure water is poured into tank A at the constant
rate oftl gal/mini The wellmixed solution from tank A is
constantly being pumped to tank B at a rate of 6 gal/rain,
and the solution in tank B is constantly being pumped to
tank A at the rate on gal/min The solution in tank B also
exits the tank at the rate of 4 gal/mini Fl (3 U R E 61.1 5 Two~tank arrangement fer
Problem 52.. (a) Find the amount of salt in each tank at any time. (b) Draw graphs to show how the salt level in each tank
changes with {espect to time. (c) Does the amount of salt in tank B ever exceed that in
tank A? (d) What is the long—term behavior in each tank? 5.3. Muitiple Compartment Mixing II Repeat Problem .52, but change the initial volume in tank A to 150 gal. 372 Chapter 6 Linear Systems of Differential Equations 54. Mixing and Homogeneity Why is the linear system that
models the arrangement in Problem 52 homogeneous?
Change the problem statement as simply as possible to
keep the same matrix of coefﬁcients A in the system for
the new probiem a = Ai+i(:), where in) = [3] U!
U: . Aquatic Compartment Model A simple three
compartment modei that describes nutrients in a food
chain has been studied by M.‘ R Cullen.1 (See Fig, 6116.)
For exampie, the constant £131 m 0.04 alongside the arrow
connecting compartment 1 (phytoplankton) to compare
ment 3 (zooplankton) means that at any given time, nu
trients pass from the phytoplankton compartment to the
zooplankton compartment at the rate of 0.04m per hour.
Find the linear system i’ = Ai that describes the amount
of nutrients in each compartment“ O D4/hr
(grazing) 0 06/hr
(respiration) 6 .Oﬁlhr
(excretion) i: I G U R E 6 . 2 .1 6 Aquatic compartment model for
Probiem 55‘ Electrical Circuits Use Kirchoﬁ‘"s Laws to detennine a ho~
mogeneous linear 2 x 2 system that models the circuits in
Problems 56 and 57. The input voltage V(r) = 0 fort Z 0. 6.3 56. Determine the genera} sotntions for the currents 1;, 12, and
13 if R; = R3 = R3 = 4 ohms and L; m L; m 2 hertries. 5'7. Find general soiutions for the currents I 1, 13, and I3, if
R1 :4 ohms. R3 = 6 ohms, L, m i henry, L: w 2 henries. 58. Suggested Journal Entry Suppose that A is a 3 x 3 matrix
with three distinct eigenvatues. What kinds of tong—term
behavior (both as r > co and as I —> —oo) are possibie
for solutions of the system 32’ m Ait', according to various
possible combinations of signs of the eigenvalnes‘? What
kinds of threedimensional geometry might be associated
with these various cases? Linear Systems with Nonreal Eigenvalues S YN OPS is: We construct expiicit solutions for homogeneous linear systems with
constant coefficients in cases for which the eigenvalues are nonreai. We examine
their portraits in the phase plane for 2 x 2 systems Complex Building Blocks
In seeking solutions of the 2 x 2 linear system of differential equations in the previous section, we substituted ii ..—, e 52’ = Aft (1) “i? into the equation and found that we must have a scalar and nonzero vector i} such that (A ~ AI)? 2 5, (2) iAtiapted from M R Culien. Mathematics for the Biosciences (PWS Pabiishers, 1933). ...
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 Fall '07
 RickRugangYe

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