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Spring 2018 MATH 307 Final Exam
90 pts total
Name:
I, (signature), will not discuss the content of the exam with
anyone until noon of Thursday, June 7.
Instruction:
• Nothing but writing utensils and a double side 4in ⇥ 6in notecard are allowed.
• Use the provided Table of Laplace Transforms.
• Unless otherwise specified, you must show work to receive full credit.
1
Heather
km
2
1. (13pts) Consider the initial value problem
dy
dx=
2x+ 1
y � 1, y(0) = �1.
Find y(1).
Y 1 dy 12 1 dx
y 2tx c
y zy 2 2 2 1 B
9103 1 3 C
YA2 54 22274123
91 7 1 f4t2 x
Yoo I Yuk l 4 722 4
YiD I 252
3
2. (15pts) Newton’s law of cooling states that the rate of change of temperature of an
object in a surrounding medium is proportional to the di↵erence of the temperature of
the medium and the temperature of the object.
Suppose a metal bar, initially at temperature T (0) = 40 degrees Celsius, is placed in
a room which is held at the constant temperature of 20 degrees Celsius. One minute
later the bar has cooled to 30 degrees. Find T (t) for all time t > 0.
Hint: first write the di↵erential equation that models the temperature (in degrees Cel-
sius) as a function of time (in minutes). Start by calling the constant of proportionality
k. Solving the initial value problem to obtain the temperature as a function of k and t.Then use the observed temperature after one minute to solve for k.
ydoesn't matter if youdidn't put a sign here
dT in the beginning You'll just endup findingat k T 20 anegative value for k later on
DT Kott720
en17201 Kttc
720 A e Kt
F Lot AettTlo 40 40 20 t A A 20
TH 20 t 205kt
Tl 1 30 20 t 20 e K
eK
for can write
TAI 20 20 e Ktk Ln I ln 2 20 20 e k t
TH 20 zoeten2 t
20 20 Est
4
3. (13pts) Consider the di↵erential equation
y00 � 2y0 + 2y = tet cos t+ et + sin t+ 1.
Write down the form of a particular solution (i.e. the Ansatz), no need to find the
constants.
r2 2 r 2 O
r 2 I2 I i
Ansatz
Ypitsthat t B etcost Ctt D et sin t t Eet t Foos t t Gs ut t H
5
4. (15pts) You have a spring in a damping media, but you don’t know the spring
constant nor the damping constant. To find that out, you decide to attach a mass of
1 kg to the spring and plot the motion of this unforced damped spring-mass system.
The graph below is a plot of the displacement of the mass at any time t. Write down
the di↵erential equation governing its motion.
Note: you should write down actual (estimated) numbers based on what you gather from
the graph, not just a symbolic equation.
The next page is blank in case you need more space to work.
free vibration wl dampingMel
Y t 8 y t Ky O
char eq I 18 I i 542 2 2µ
Soln yet C e Etcos ut t G e E sinMt
A e It coscut 8
Can immediately read off from the picture that1 2 8 0 quasiperiod 2 µ it
continue next page
6
(Extra space in case you need it.)
so ya 2e It cos Et
To figure out 8IG 8
1 2 e cos1212 2 ereadoffthegraph
e Ir Ln I ln 2
8 lnl2l I 0.69
To figure out KI µ 4k
2
4T 4k lnk2
k Tilt 0 29.99
Y tlnl4y't T2tµyµµ
7
5. (12pts) Find the Laplace transform
L{2y00(t)� ty0(t) + 3y(t)},in terms of Y (s) = L{y} and Y 0
(s), where y(0) = 2 and y0(0) = �1.
Hint: use #16 in the table of Laplace transforms.
L 2g It ty.lt 3ylts
2L y t L t 3 LEY
2 5Tls sylo ylo dasL y 3Yls
using2 5Tls i t da sys y.gs
3Tls
25Yes 4 s 2 t ST's t Tls 3Yls
SY's t 25 4 Tls 45 2
8
6. Consider the mechanical vibration modeled by the initial value problem:
y00(t) + ⇡2y(t) = ⇡�(t� 1.5), y(0) = �1, y0(0) = 0.
[a] (14pts) Find the solution to this initial value problem.
Hint: use Laplace transform.
52 sy t Te2Yds I el o
54 Td Is s t et 55
lossTls Tj 7
Tls e s s z
Hls
htt L Hls sin It
Ytt L Yes viisits htt 1.5 Coset
ycts Uuslt sin T t 3z co5f4
coslat 0 Etc 1.5
sin Elt E costa t II snd
For part bMethod1 Methodi
costt sin X 3 CO'S XviHt sin Tilt
Zso ylt cos t Ost l s
costa cosents is
emand cancelafter tH5
9
6 (continue)
[b] (8pts) Graph the solution found in part [a].
Hint: if you couldn’t solve part [a], you can still attempt to graph the solution as much
as possible to earn partial credits.