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I.D. number
Math 22SA-l
Practice Final Exam
(Mostry
*" 3ffiff""1*1,** zoo4)
This examis closed-bookandclosed-note.You may usea scientific calculator,but not one which is
capableof graphingor of solving differcntial or linear algebraequations.IntegraltablesandLaplace
TransformTablesarc includedwith this eftlm. In order to reeeiyefull or prrdal crcdlt on any
pnoblem,you muct show ell of your wor* andjusify your comlusions. This examcountsfor 3A%of
your cour$egrade. It hasbeenwritten so that thereare200 pointspossible,however,andthe point
valuesfor eachproblemare indicatedin the right-handmargin. Good Luckl
l) Considerthe differential equation
P' (t ; = -P 2 + 7 P -rc
lai Find the equilibrium solutionsof this differential equation.
P'lfl = - (faZf * rd
=- (P-s)e-t
(2 points)
? *2,5
(fh"s.
1,*
.t
5
z
o
z -olg
5 o
t -rl
3 z
+
c
IL
4
o
lc) Classifytheequilibriumsolutionsasbeingstableor unstable.
P = 2 is .^'sl4k
F =5 i:, s{oLtr (au.^t tf'-tt
(4 points)
)
ld) Use separationof variablesto solvethe initial valueproblem
P' (g = -P 2 + ' 1 P -1 0
&0)=4
forP(t).
(15 points)
?2:
-*' e-tL
P-z
?-s, (r-4(-tc-tb)
(t+;at{)=
5+e-3t
s,
@t . o :
d r r il., ? ( tta S * L ^o
rbt ' I " 4r/
7L
p - : f-=v -{
h
{-L
le) AOO(or higltligh$ the graphof your solutionto (1c) to the slopefield picture in (lb). Verify below
that your formula for P(t) givesyou the limit predictedby the slopefield, asP approachesinfinity.
(5 points)
[iL-ao
5 * e-\E
I r!6 . 1 t
-?5
l+ o
4
r (Jl^t{3 {'.-t-
,"^4{- grcL-c a^-
Y"* 1.
?al A small motorboatwith passengerhastotal massof 150kilograms,and its motor is able t,
of boatvelocity. UseNewton'slaw to explainwhy (while the mo\
differential equation
I
lv
u'(t)=; -
wvt = r.r} [^*s
30
(5 points)
ls o v t ; 5 o + b
.50 vt 2
I)
z -5(
So-Sv
sc r', t- iort = thn
2c) Solvethe IMTIAL VALUEPROBLEM forthe boat'svelocity,assumingtheboatstartsat rest.
You areto solve it TWO of the following FOUR different ways,your choice.-your choicesare
(I) FIRST ORDER LINEAR (asin Chaptert)
(rI) SEPARABLE
(IlI) NTH ORDER LINEAR (with N=1) (as in Chapter5, with particularand homogeneous
partsto the
solution)
(rv) LAPLACE TRANSFORM.
(20 points)
If you makemultiple anemptsyou must indicateclearly which two areto countfor grading.
-L+
3
c- r
=
{ v ' * fo u i
lvrot,o
f,r ("'*
e
a
i,
,")
=s
v s lO
v( o ) :
dY: - (u-r)
*.
d+
-4g. -fo4t
v-lo
e3
tr , ) r_t 3 .i't
.+r It:
.
,*t,
T)
-t
=
i€
+cj
to e
ir
J-&
3tl '
*to + c 4
(=-tD
tt
v: to * C.-
II!a'!{:^J
v ( o ) c O { C= - lD
uG;i,u
+u
g
,r'+ tou = J.t
v =rp * vrt
/
Jf
-{o*f v(s1=
l5
fut sV(s)
Ugtt:r+{o.g
,*'" j: g,o
"
+nYe=
a
'?'=o
vrtrfov=o* c .L
gaw
5 o V: tO +
V(e)=*(fur)
i3'(+- iE.)
Vti)
703
4
V[e)(s*fi)=t +
? rq_lg
TT.'
c;?r
s
s +'6o
v [tl
vto\ : O * 1 ( = -l o
rod *t
2d) How long does it take the boat to accelerate from rest to half of its terminal velocity?
(5 point$)
S nt"e-vtt): fto:5
t
lo-rbitt
=5
5'lodf't
.5'{t" t
,o^ts) = -!rb
+ = -lo !*'('s)
= 3O !t-z( % 2o.8 v.)
3) Considerthis initial valueproblem,whichcould ariseasa springproblemwith forcedoscillations:
x' '(t>+4 x=3 co{z t)
x{0)= 1
x ' (0 ) = t
3a) If this vrasmodelinga singlemassand2 sp$nsconfieurationasshownbelow,and if the massm
yas I kg, what value (AND UNIT$ffffi-guEdnrt*t
k would leadto the differential equationabove?
(4Points)
F}*ti*.5-f
H
yr**ll:
* ta1
'a -Zh
lr':,[:
lt)
- kx npqf
* + Frt)
,.tt v z[x
.F
v/v'-
3b) Solvethe initial value problemabove,USING LAPLACE TRANSFORMS.(Hint: usethe tableyou
havebeenprovided with to do the inverseLaplacetransformations.)
(16points)
x " + *x = 3*s zt
.2f X t r \ - i- 2 + + X (s )=J
5
st+tl
XI'r(*++)=tfr* + s +2*
Xk)=dh.-fo"fu
sin zb $ b sa L + str tl.t
\r+x l , t ) = 3 t
t
t\
-#-
,fe,bh
V"4
3c) What hsppensto th€ solutionto tris problemasr increases?\I/hat physical phenomenon is
exhibitedin this forcedoscillationproblem?
(5 points)
l,
\r-escr^a-r^<^c- W"
f"*rrl. of *L^-(
A"l k*)
, auJ C{J^.
h**
t.". "*if"Lot {vohrrrS,'**b ir"t'r'^'t--
4) Considerthe homogeneous
linear differential equationfor a function x(t):
x, , ,( t ) + 2 x , ' (t )+ 5 x ' (t ) = 0
4a) What is the dimensionof the solutionspaceto this differcntial equation?
(5 points)
\^A
4b) Find a basisfor the solutionsto this DE.
(10 points)
1.
y (r l = l ' t r r 2 r L + 5 r ' - o
= r ( *+2..k 5)
/
\
\,
4=o\o.
Q+,,1-*
ts:o
t'.J"=-?
/
{,I t)'
r t\? t2 i
I
ts=-\tu
5o
v=-zl'lF
=-t{";
e(-t+z*)t = i+ Gs rL + ;sirnrt)
L,, tL^rrL, /t *rt\
is
ct b oh).
4c) Find the generalsolution to the differential equation
x'' '(t) + 2 x ' ' (t )+ 5 x ' (t ) = l$
xH = (r+ 6e-Lt*srL+ge't s;nrb
(10 points)
XP= Ct
't
I
tr*,.r,,t C s'f"+ 16'
+
\'= c
f,=
t'n','= o
s" L*? * oro tq C-rl?
L'z
\* !E
K :2 + c ,+
tE
-Lt
CrC 6SLL * qe 5,i^?-b
5) Considerthe matrix A definedby
^[:i]
5a) Find the characteristic polynomial and factor it to find the eigenvalues of A. (Hint, the eigenvalues
you get should be = 0, 4, -5; your job is to derive these values.)
g*er-rl o -o+r
lA+r1=
3
li-^J^ I
l-6-r|
[o
6 - z ' l-[
1 6 -t+ l
=tr-^)Fc-r)(-z-l)
+ tL
(5 points)
r -6-),I
6
L I
;-(X.+0(Xr C)(X.+al+ re
' - (f+rltt) tl'+z) tlz s -fSttltr
,'=L
I
5b) An eigenvoctorfor l, = 0 is given by
zo),'rz+ 17
: - ( f + 11. + a{
3 -l ( A1+12.+e,o)
5 -X(XtqXr+lJ
h-b X=O,-1.-5 t/
Find eigenvectors for the other two eigenvalues, to get a basis for R^3 made out of eigenvector$ of A.
7\=-'l
Ito
l-l
07-
3
I
-L0
o
6L
ur- t - Zo
oC z
jL+t,
o6
l
+r+*,
Ra
7-
o
-Lo
o
o
D
3r
o oo
o
v,. niF
o
q3
t
- \o
,[1lj
Ifil,Lil
r -l o
o L t
o oo
n"Fq ;
'Hl
v=r
V3* *
vz=-th
z
o
'qo"q o +L
o 63
o
o
(20 points)
X,=-5
J
V
ril'
Oo
? ,L
o
v3=t
urz-lb
V r r - t{ b
i"t
.,
v:,
til
fil
6) Considerthe following three-tankconfiguration. In tank one thereis uniformly mixed volumeof 6O
gallons,andx(t) poundsof saltsolute. [n tank two thereis mixedvolumeof l0 gallonsandy(t) pounds
of salt. In tank threethereis 30 gallonsof liquid andz(t) poundsof salt. Waterus pumpedslowly from
tank one to tank two. frorn tank two to tank three,andfrom tank threebackto tank one,andall ratesare
60 gallonsper hour.
xttl
trt)
* !a?
6a) Model this tank configuration, to arrive at the first order system of differential equations
dx,
X.t= t 'qi- % . o= F 3 - L o +
3o
6o
q 2 ? -x
dt
gdt
1'r6o#-r"e
{:
dz
dt
=n-b?
+za
5o x/=-(
2 ' = 6 o5 - c o!
lo
30
J'= x -tY
= nA_r,
Z'=
:lLll
t* *"o
(5 points)
)w
63 4z
6b) Assumethar at time t=0 thereare40 poundsof salt in tank l, 20 poundsin tank 2 and40 poundsin
tank 3. Solvethe initial valueproblemin this case. Note that you havealreadyfound the eigenvectors
andeigenvaluesfor thematrixwhich appeaniin this system,in problem5.
(15points)
"H],
c,
.,;*'ll]
'f
t
*5 [i]
xFtt)
e[ ,c .o:
G -2 -l
4o
-l
LO
t -t
33
81
-6htq
_ln
4
r -t-t
o +5
o6q
L
{lt>
to
zo
?
nsu
tro
lo -'lo
I
o
o
lo
ol
oo
to
3o
-fo
$o
=ril
liil
,,[:]
-{o
i*,.o -,,r-'ril
;i]
Sr , S ,4i'r*
'l
6c) What happensto the salt arrcunts in eachtank ast approachesinfinity? Why would you get the
sameanswerif the initial 100poundsof salt weredistributedin different proporrionsto the threetanks?
Explain.
(5 points)
= rof tl - {^n"]
ti^ ir.+1
#.*
ll I
lto I
LsJ Lr"J
-H.^'s,^,,'ll h.( 'H^.e,r:Jrrdv {xa"2 tv}
[cc^a-gr [r,'n^{r^1 crtt'-"{t'.ln'o^.
i* at t 3
r^'t t laq
"d,/
So ao.n^.r^"( a^'tt l.l
(^ r,oerts tpL0J"lf)
to,^lar,
{Dfu'& u'r[n*r"
g-1o.'l"a^^,(.
7) Considerthe following configurationof two massesheld togetherwith a singtespring,on a
frictionless table. The mirr on tie tet is 2 kilograms,the one on the right is 6 kilogfams- The Hooke's
from restpositionto the right, asusual,
constantk is 6 Netwons/meter.Measurcpositiie displacements
indicated'
as
and usex(t) and y(t) for thesedisplacements,
F*t
t-t
-
Xtt)
*r'tt
?a) Showthat this mass-springsystemsatisfiesthe secondorder systemof differential equations
:ll;l
[:::'J=[_:
Lr < " = 6 ( j - x )
(5 poin*)
x"= - 3 r + 3 "t
6 { ' = - 6( 3- x - )
5'' r^To
Tb) Find the generalsolutionto this systemof differentialequations.Hint: This is a secondorder
syite* of twddifferential equations,so the solutionspaceis 4-dimensional.You will needto gettwo
above. Think aboutwhat you are
linearly independentsolutionsfrom eachof the matrix eigenspaces
with zeroeigenvalue.
get
eigenvector
the
you
from
can
modelingto understandthe solutions
(10 points)
i " = A*
+rO.fu.-b-U
V, st*'' t
r'os,.rt
t
*.{t ,^}t-tr,N*:€.a'[1A
(av'1-3 = **42*
' tA-rtl=
* (x+gJ
_L I
\-,;^
o
2..=
*1 j o
f
\ -r
IO
n'tll
2..-4, , *.L
D1 ]
lo
I 3 \r r
o'r11.
f [ A V= d
+1's*l(tr,(, +<"t)il ulo
i"='o = C*r.t)Ad !
i ttt'(c,
= X(r.+4)
,F
(r,ostt r q{9,1}+
+
-',.)[l]
r^tt*^ s Fh^1
sf'lt, nbihL
ItVe.Sl*t
t+-o"*V
a[ U^.s"r* s(uJ,
r^i.lL 3*r. (,
ose^'llalv
or^t cb 0
gl*rtt
na5 | a1l.
= lh4
Yst 79t ,
8) Considerth€ systemof differential equationsbelow which modelstwo populationsx(t) andy(t).
(You canthink of this asanextensionof problem(5) for thepopulationx(t), which now finds itself in
the presenceof anotherspeciesy(t).)
l a 'l
I o, l[ zr-"-rll
ll=ll
t# l t * y +x y r
(2poin'ls)
t,y-ff-1:bt
\j..3,i f,3,f{.
7ur.+n)
8a) If this wasa modelof rwo inrcractingpopulations,whichmodelwouldit be? Explain.
T":lt
ffi* [,,fl[H,..u)'
8b) Find the equilbriumsolutionsto this systemof differentialequations.There
"*}qo"lhffit:
x*o 6L z*-i=p
1tc-xL-xy=o=r(-l-x1)
{S * 15= o =J (-e*J
./
Y=o
/r.,
$oxv
lql
LO)
Fl
LoJ
^lo '
J
Gpoina)
\*"r
V
l=2
--^l
f'51
L zJ
8c) Only one of your equilbrium solufionshaspositive populationsof both species.Linearizethe
populationmodel nearthis equilbrium point. U$eyour analysisto classify which type of equilbrium
point this is.
(I0 points)
rll
r= f F *? l =
Lcrc,7J
3(s,,)=
f-t
L2
rlrrl=
L;
3
-r-: ]
L'^*^^t"L,"-
;'llt
L:1=r:
-5
- 5\
I
_^l
l?
: f+5 1.+to
4 -^ <- fi
' { ' rc ,
?o
2-
^=
4s9=rn
=-e,
1 rg 2 / a LtI:-.
t
Pt-4u-
Qt:y
\
eltv.
1L
"*'(*r$t -tn$t)i
8d) Using your work from (8c), fill in the missingpieceof thepplanephaseportrait below. Also draw
th" thg phasediagramsalongeachof the positivc x andy axes. Thsn makea p,redictionabotrtlong term
behaviorof solutionsto this populationmodel,assumingboth populationsstartout positive. (Depending
on your analysisin (Ec)yourprediction may or may nof dependon wherein the fimt quadrantttreinidal
populationv€ctoris located.)
(5 poins)
x'-?*+F+q
10
Y'* -6f+xY
I
l
t iJ
I i I
t i r'
.l
{i J
lil
/i ,/
IiI
I i I
Iit
t :t
I Iit
.1. f : t
6
l i t
. t I' I ti t
ti ,
-
a' i
v5
t
t!t ,
l :,
ti t
t
t:t
lif
tl
II
I
I
l;t
I
t:
l
I:\
tr
l :l ;
I
rl
L r;
'-'- ----':"" ""'t"
4
3
t:l
r
t
lI :
;
tY
l:
r:\
\:
\:r
\:\
i-
. i-
.- i *
*- !s
si*
c-.it-.. r--it-- r--
. -i* *i* *. iq *i**i***
i *,
s-- i t
l+.
"""1"f
G \ : r F \ -t\-
ta a
- i*.- i**i**i< - - q iq t- - !r - - s+.-iq *-!+- +-3s--*..i
*-ir- *i*
-
t i t
t i J
+ . - is - r s i\
*- :t- s:GG*:G\+ "' :!ts' .tr *:+ \G\
- i-
- - - - er - r J r r r - - . . . - i r . . . . . . - r . . . - . - r - J - . . . . r - . . \ r r r r r r r r l r r . . . - . - J . . . . L . r . . 1 . . . . . r . - a - - - - - '- - .
l.
+- :+- €:*
r i-
J
a t
a e/.-:c'
"'-':.f""""r':'--"'1"-"
(t-
; 'F\*i;G-
+- lt---
<-.-!tq..
tr--
v-.i*-
g;g
g
!r\-:th.
Gb
- "'1" " " "1----- ---
\
:a
G..: rh.
rF.-i !F-
+-: its... {-:<t-
(F .-
(ts,
. . - . . .-:..:.-:......:'..-...-.l ....- .Nw:- ' ...- .- .l......._.:...' ..".
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.tt\
It.\\.\t.til
hi raaa
* i r..
*l+-
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o
o
i';
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5
6
7
.\
89
I
[ ,^sror,["] 1=['t -t ]fnl +"
Lu'l [z , lL,\
10