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MATH 267, Final Exam , ISU, Fall 2020
Student name: Section:
Instructions:
• This exam has 10 problems, each worth 10 points.
• Print this exam or use your notebook paper to write solutions. Combine your solutionpages to a single pdf and name it as final yourname. This first page of the exam with thesigned pledge (see below) should be the first page of your final exam solution. Follow theFInal exam link in Canvas to submit it. Download and doublecheck your solution to seeif it is the correct file.
• Discussion among students about exam problems in piazza common forum, in-personmeeting, email, canvas discussion board, or online in websites like chegg is strictly pro-hibited. Reach out to your instructor via email or private message in piazza if you haveany questions about the exam.
• Your exam will be graded based on your step by step work, explanation, and mathemat-ical correctness. A solution that only has the correct answer will get a score of 0. Unlessotherwise stated, you may use any valid method to solve these problems.
• Use all the resources available for this class, including the internet and calculator. We useTurnitin software to find out if your work is plagiarized or copied from somewhere. Anyacademic misconduct will be reported to the dean of students office immediately.
• Laplace Transform table is attached with this exam.
• Absolutely no Chegg! Because of the recent use of Chegg by students, ISU officials
are in contact with chegg.com to determine who is violating the ISU academic integrity
policy. Severe consequences are in order who violated the policy after signing the
pledge. Chegg data will also come up when the company you are applying for a job
looks up your online presence. Please do not jeopardize your future by using Chegg.
• Read the following pledge, sign and date it. If you do not have a printer, copy the pledge,sign and date it. This signed page should be the first page of your exam 3 solution.
“I pledge on my honor that I have not violated the Iowa State University Academic
Integrity Code while taking this exam.”
Signature.................................... Date........................
1. Consider the differential equation y 0 = y2 - y- 2.
(a) Find all critical points and use a phase diagram to determine their stability.
(b) Use any method to find the solution that passes through the point (0, 0).
yeol I
0yay a o H 2
Tty
4in14yd y C 1 Ct
y o yt t it
io
dfa2 eyal DI da
H's t.lda.fi2
dy 3dglue
Int Infy y Inayaton the
Inc EI encode
2. Suppose the population P(t) of an ant colony grows at a rate proportional to the popula-tion. Suppose that the initial population doubles in 2 weeks.
a) How long will it take for the population to become 10 times the initial population?
b) If the initial population is P0, write down an initial value problem for P(t).
Po pcoi
pH 9ft 2 Pcts 2P
o Poem em 2 72
qf petsb Post µpµpgekt
Pit PEI mo Eze 1m02
d Pipt 2fL
3. Solve the following initial value problem.
xyy 0 = 4x2 - y2, y(1) = 1.
y FLI ee
s
yhe i
F EmHomogenous
egosfistader
y Un y V t se v
v xu4En IL
ft n v r Hz v v
se v 4220 4 12
data2 ftp.mdalnfe
am
142 Indu
dat Ya da o
Integrate MuInforytalan the
lnLC2vy.se Inc
v2 sell cY Vn
f Ina4 c
222 5 se C
ya 1 7 2 i c c I
2242 al I re
2u2_y2 I
ylra izz
4k
s
4. Solve the IVP:y 00 - 5y 0 + 6y =
2
1+ ex; y(0) = 0, y 0(0) = 1
This is a long problem
Good for foraefive butnot good
for the exam
5. A mass of 3 kg is attached to the end of the spring that is stretched 20 cm by a force of 15N.It is set in motion with initial position x0 = 0 and initial velocity v0 = -10m/s. Findthe equation of motion. Also find the amplitude, period, and the frequency of the motion.
ur
Hooke'shw f E I o fsn ma t Koe o
32 t 752 0
se t 25N0 seco so uhD 10
Sf usc cossttqsnnstfmstux.ws0 7
st
se 512 Cosst
N'cos to I o 5Cz 92 2
HH2
NANAacts fi
Amp 2 St i it2
periodfoequey period
6. Solve the following system of ODEs using eigenvalue method.
x 0 = -4x+ 8y+ 8z
y 0 = -4x+ 4y+ 2z
z 0 = 2zIlx Hal ftp.fi.IEHH
X Ax where A
gonna I Io
et
Cfa a cu a323 0
16 4 7 147 72 36
7416o
7 3 7 Iai
a
t.io ll l lo7G U 1842 843 0
y 4 Ui 1242 1243 0
My 1243 0 62 43
U 20 Uzi 43
7 2 U Y
x ni
Iii Hit
3rd 2 4103 0 03 0
2nd 40 the 44021203 0
O
40 4C c Iz
His f 402
a ai IiiLuite
7. Find the general solution using the eigenvalue method:
dx
dt=
2
4-2 1 2-1 -4 10 0 -3
3
5 x
eigenvalues 1 If a 3 f o
f 3 x f 2 a C 4 x IT o
7467 18 11 0
746 At 9 o
X 3 f 32
0 7 3 330eigenvector E III Io
tutti fiddle fitGenaaeeskiifio.iooHtg.I fit
to is is Eel
Generalizedeigenvector 2
toolmenu
t.io w fsfEdSolutions
Est ftXs E If t f iestf.tt
xz e3tLfItea.tf9oIttfggD
e3tfIytzttYGeneral Solh is
Xi C X 19 21 5 3
8. Use either the definition of Laplace transform or the Laplace Transform table to findL{f(t)} for
f(t) =
8><
>:
e2-t if t < 2
t if 2 6 t < 3
et-3 if t > 3
I Ult zUct 2 Ult 37
Uct 37
fit eti ult 2B ft UEy uit733
yet 3 Uit 3z te t Ust 4 Et e T tu IIet7J
E
eEt lect 216204 2 e
1 ET Iiit 3 Eet ftDB
i fEa gainsuit a felt e
E Et
sEes ie ii
9. Use Laplace Transform to solve the initial value problem
9y 00 + y = sin✓t
3
◆+ �
✓t-
1
3
◆- �
⇣t-
⇡
3
⌘, y(0) = 0, y 0(0) = 0
where � is the Dirac delta function.
Sin att fS2ta 2
ie
f f gy tyf LSmtz t f I t 437 Jct Ib
g f s Y It y 3 g
t E es
Y f
x i seei
ie
Yes Ia E u I FEI
H r3 E ez
apply a is i E iit
our at it oil eIIu
3 t tU't is 3 SmC
38m17 gUl t 931 3 Snf3
73
y Iz EaSmTz Ijf tasty
1 zUI t T Sn 3 izU't Ms
Sinftp.i
niiI uO
10. Derive a recurrence relation giving cn, n > 2 for the following differential equation. Usethe recurrence relation and the initial values to determine cn. Finally use cn to find theseries solution of the differential equation.
y 00 - y 0 - 2y = 0,y(0) = 0,y 0(0) = 3
.
I n EEnau.i
nIanennenanQnI.ncn zI.oc
o
n n 12 Ntn 11 y
FIm.fntyfntycmzInO nI.olntilfnt7O_nIoHn
000oa.Io
Untyenakan Hyla4 xn o
nth Cn nfhDG
cn ii HI.inniiiin.iRecumoneezelahm
O
cnu h tIa
co y 3 Cd d
co c
Cz yr cotfa.ec a4czse3 cqa4
9 144,7mFn o Cz Iz t 4 In i Cz Iz t 4 L j 133 32
n 2ca r Ee t 9 2
324 3 tIs g Is
n 3 as tLgle
Table of Laplace Transforms.
f(t) L[f(t)] = F (s)
11
s(1)
eatf(t) F (s� a) (2)
U(t� a)e�as
s(3)
f(t� a)U(t� a) e�asF (s) (4)
g(t)U(t� a) e�asL{g(t+ a)} (5)
�(t� a) e�as (6)
tnf(t) (�1)ndnF (s)
dsn(7)
x0(t) sX(s)� x(0) (8)
x00(t) s2X(s)� sx(0)� x0(0) (9)
x(3)(t) s3X(s)� s2x(0)� sx0(0)� x00(0) (10)
f ⇤ g(t) F (s)G(s) (11)
tn (n = 0, 1, 2, . . . )n!
sn+1(12)
sin ata
s2 + a2(13)
cos ats
s2 + a2(14)
eat1
s� a(15)
f(t) L[f(t)] = F (s)
teat1
(s� a)2(16)
tneatn!
(s� a)n+1(17)
eat sin ktk
(s� a)2 + k2(18)
eat cos kts� a
(s� a)2 + k2(19)
eat sinh ktk
(s� a)2 � k2(20)
eat cosh kts� a
(s� a)2 � k2(21)
t sin at2sa
(s2 + a2)2(22)
t cos ats2 � a2
(s2 + a2)2(23)
sinh ata
s2 � a2(24)
cosh ats
s2 � a2(25)
t sinh at2as
(s2 � a2)2(26)
t cosh ats2 + a2
(s2 � a2)2(27)
U
TOG
C u
v