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.