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(MATH150)2002_spring_final_exam_solution.pdf
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Math150,L3,L4,FinalExam,Spring2002

Date:24May2002Time:4:30p.m.|6:30p.m.Venue:G017
Name StudentNumber TutorialSection Score
1.Atankwithacapacityof800galoriginallycontains300galofwaterwith200 lbofsaltinsolution.Watercontaining2lbofsaltpergallonisenteringatarateof5 gal/min,andthemixtureisallowedto.owoutofthetankatarateof2gal/min.Find theamountofsaltinthetankwhenthewaterreachesthreequarterofthecapacityof thetank(10pts).
Solution.Thechangerateofthevolumeofthesolutioninthetankis:
5;2.3gal/min:
Threequarterofthecapacityofthetankis:
3
.800.600gal:
4 Thetimewhenthewaterreachesthreequarterofthecapacityofthetankis:
600;300 .100minutes:
3 Thevolumeofthewateratanytimepriortotheover.owingis:
300+3t:
LetQ(t)betheamountofsaltinthetankatanytimepriortotheover.owing.Atsuch timet,theconcentrationofthesolutionis:
Q(t):
300+3t
Thusweobtainthefollowingequation:
dQQ 2Q
.2.5;.2.10;:
dt300+3t300+3t
dQ2Q
+.10:
dt300+3t
dtchoose2n
l(300+3t)3
200+3t
e R2 .e.(300+3t)23:
1
Hence
R
10(300+3t)23dt+c
Q(t)..2(300+3t)+c(300+3t);23: (300+3t)23SinceQ(0).200,wehave:
600+300;23 c.200.)c.;400.30023: 23
100
Q(t).600+6t;400:
100+t
Whent.100, 400
Q(100).1200;400.2;23.1200;plb: 34 2.UseLaplacetransformto.ndthesolutionofthefollowinginitialvalueproblem (8pts)
000
y;4y+5y.4u3(t).y(0).2.y0(0).;2: Solution.Lety(t)bethesolution.Set Y(s).Lfy(t)g: ApplyLaplacetransformtotheoriginalequation:
;3s
es 2Y(s);2s+2;4(sY(s);2)+5Y(s).4: s
;3s
4e
.)(s 2;4s+5)Y(s).2s;10+ : s
2s;104e;3s
Y(s).+
s2;4s+5s(s2;4s+5)
;3s .
2s;104e1s;4
.+;
(s;2)2+15ss2;4s+5
;3s .
2(s;2)64e1s;22
.;+;+:
(s;2)2+1(s;2)2+15s(s;2)2+1(s;2)2+1 4u3(t)
y(t).2e 2t(cost;3sint)+(1+e2(t;3)(cost;2sint)):
5
3.Solvethefollowinginitialvalueproblem(8pts)

00
y+4y..(t;2)+tant2.y(0).1.y0(0).;2: Solution.Lety(t)bethesolution.Set Y(s).Lfy(t)g.G(s).Lftant2 g: ApplyLaplacetransformtotheoriginalequation: s 2Y(s);s+2+4Y(s).e;2s+G(s).) 2
(s 2+4)Y(s).s;2+e;2s+G(s).)
;2s
s2eG(s)
Y(s).;++:
2222
s+4s+4s+4s+4 Thus
11 Zt
y(t).cos2t;sin2t+u2(t)sin2(t;2)+ tan.2sin2(t;.)d.:
22
0
4.Findthegeneralsolutionofthesystem(8pts)

.
2;5csct.
0
~x.~x+..t..:
1;2sect2
Solution.Let.
2;5

A.:
1;2
Thenthecharacteristicpolynomialis
.

2;.;5
jA;.Ij..(2;.)(;2;.)+5..2;4+5..2+1:

1;2;.
Sowehaveeigenvalue.i.
Since .
2;i;5

A;iI..
1;2;i
~
theequation(A;iI).~.0foreigenvectorsyieldsaneigenvector
.
2+i21
.+i:
110
Sowehaveafundamentalsetofsolutions:
.
212cost;sint
cost;sint..
10cost
.
212sint+cost
sint+cost..
10sint
Thuswehaveafundamentalmatrix
.
2cost;sint2sint+cost
.. :
costsint
Moreover, j.(t)j.;1:
.
;sint2sint+cost .;1. :
costsint;2cost
So
.
;sint2sint+costcsct2tant
.;1(t)~g(t). .:
costsint;2costsectcott+tant;2
3
Wecalculate
R. . Z
2tantdt ;2ln(;cost)
choose
R
.;1(t)~g(t)dt.. :
(cott+tant;2)dtlnsint;ln(;cost);2t
Hence
.
Z
~x..(t).;1(t)~g(t)dt+~c
.
2cos