OUTLINE OF SOLUTIONS TO SELECTED EXERCISES

FORSIAl") 0lQs.5'lc..s In ;+1f:>~~J't1~(IIV.-h'aJ 2-1 L 19<12»

MATHE~~TICAL MODELS:Mechanical Vibrations, Population Dynamics, and Traffic Flow

by

Richard Haberman

The following solutions are included:

2.(1,2); 3.1; 4.(1,2,3); 5.(1,2,3,7,8); 7(1,2); 8.(1,2);9. (3,4); 10.(5,10); 12.(1,5); 13.(1,3,8); 14.(1,2,3,6,8,

9,10,11,12,13); 18. (1,2,3,4,5,8,9); 19. (1,7);21.(1,3); 22.(1,3); 23.(1,3); 24.(1,3,6); 25.1; 26.(1,2,6,14);28.(1,5); 32.(3,4,5,6); 33.1; 34.(1,4,7,8,10); 35.1; 37.4;38.(1,2); 39.4; 40.(3,7); 42.(2,6,10,13,19); 44.3; 46.4;49.2; 50.(1,2,3,4,5,6); 51.2; 54.(1,2,4,5,11,12); 57.(1,2,3,6)59.(1,2); 60.(1,2,3); 61.(1,3); 63.(1,2,6,8); 65.(1,3,5);67.(1,2,3); 68.2; 69.1; 71(1,2,4,8); 73.(1,5,6,7); 74.3;77.(1,3,5,6); 78.(1,2,3,5,10); 79.3; 80.1; 82.(1,2,13)

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tt-J+C(}r~-b"d- <a'cJ.ds

\1'\'}l ~ - ~-t ,..C.1"-5mY\t:

jn.-h"l~ V~ >/, ,'" r 1r ~ - ~i + 11-:]

~n()rild (~r -rte v~1~"r\e""t)-tfc pDSS'IIo',I,-to

0) ~IT",~ 6ff , -M:e- %-b~, -<;tz, pS

L..,I,e", 'ftc= 0 , I.e. Ii: = ,~vVI>J(we. uJa.d:: 1:0 lHcuJ lJ.>~eAe ?\Las~ I~

. )6..1:.~-t -hvne - YieJ pcj cblJ{\.)

I /e~ ["Z',;b"(t &,,"" Clf'p~ I~O tMhcJ) ~.utI"", )("'~X -= ~ L 1:2- +v-tJ,1')j ~

At.., SThPPIf\(j t)I'he... >< ~rt\ vOh V

Fo.\IS 0-[ 11 1t~ dtJ fr:\~ce ~Ved~ ~~ \i.J~Ie. D\V,l. N~ ".> \-\ ~ r \ (--. ZK() nL~ 'Ii,}"'" -

vY)

10./0 (a.)

\2. s'SDlu.t.Q'Y\ 5 ul'\der-c:bVI\.{Jed~Pfl{ld-muss S'-jJlen.v

-CVZM / ~ '~ )X = e., l C1 <J:.~L.U1- +- C/..Slv--L.v~

IV\\-ti c--D C-(;~d \~ ~ -S

X Ib):::: 0 (i)\.j)lte~ 0 = C.I

jll~J= Vo (HeeJ C~~)

t l - J,X -,--t;{M / -I- ~

WI'" CI -- ~ ) Jct: = CJ.. e" \.LUctS(;J'- - ,iJ)1Juv

1V11+1U.!(if . Vc -= Cj".W

~(e-~re ~x- - Vo - ct.;:.fY\" ..~-.

- - e $1.1 W t.-. W

G.W'--p1.1u.de (:if 0.( c d I~-b 13'v\ :

Vb - L-t h.VtIWe

12.r; m d~~1! :=: - kx - c--dX/Jt. .~ D C{)~ ~I()i:

~) c. =- 0, vn~~2- -::: - lx + b CD.S W;t:

--teld-l' a Jcvv .s,<:)ltA--b oY\.,

Wo i= ~ B

XA> = I< L U, ~ uJDt-r -mwc

u\ .= ~ ~/rY1 x - -6 . --L-t - 2mwo 1: SlruWoJ-.

(1)

jeV\€-tcJ1 Sotl.A,-L~

X~ Xi + LI CbS wi t c).s-,v\'w -t j urf€/te. W =-rYfYL

c -#- C) X -{J = v< 'C{) s uJ01- t- ~ .S'I f\., U)o1:-

Co.SuJDl:. : - fY1w~ 0( t k a( t- cU1of.>-= 6,

5/1\ w()-t. ~ -(v)UJ~~ t- k ~ - ~wQo( = 0. '6 l k- fV1w; )

cI. == - I -).z.

c.1..wc""+ Lk-m""'~

-50\ve :

\:) c.. \..v'C'

\ - c,;'UJ~+(k-(J1UJ~Jz

X-.' . c - c tj z()') . (-L j... )- X-ft- k e s,vv u.d-'+ 4-'0\. ----

C-x-pcFleV)-h~ ~d'

{'1-k c

u-rfe{ e w':::; m-,;;;; 2-/

-+r lCU3° 1:uY\e X .~ 4 ;:. A ~if) {()Jot + Be) (~erv..-

ampl,tMe- A = fo<2+rt = ~2, . c:'2.w ,,2+ (K-VVIl..ut)

(c.).p hCts;e

L

j -I k-t'YIWo. L -I!)(... J..e .:::: 10 rL - ~ l.Ufl,

0 f C-wc

fV\ k (1- W:\1'1/k)2- J .~~"-'~f CZ -<-<.

Q~Uine,i 05 1aV1.-I 1.5 lcu.5 e- . T~eAef.r0

90'~ ~11/:2.. 'If WOLL. vn/k

l -11/:2 'T Wt.2 '7 ¥'h/k

+rcl~ d ~

PJ CoS wet

re.8 pJY1 <:>e i

e:~~irvLCJoi_+ec~ '

+lW~ + (k-'VlI-V~)2.

iT w: L v~ \ ~ IVl(uJ~t i-1T-/2.) -:::..Q)S w;;i. If\... ~cw Q...J

If Wo1. '? Vhl<-} SIT) (WD -I: -"/2-) := -CbS L0z;t (80 a oul \>~ ~~

[rte. SWn e- r-<%JuIi. c.A',sfs -If ~"'- (Ja-'Ltl~

<;blcd;Joyv em\1 c>f YCVL.1:-~ 115 6N<.Q~Jed j \, e 'J c=-o]C9) {'o;ponse= - \

-{ c/w; + (k-v'YluJCL)2-

-rfe- ~o'y) se- WIt{ r-&Jfl€'~ -t-IJW~L

2.. - k ,c~

lut) - ' m - 2t11L

. .

(\1d.,X IVT\Id I0(j-leJJ~

r5 c/- .<:L:. -4 VV\k j -if ~ h.-2.. '

Wo ~~YI1

(, ) I

4( -?

)-Gt4~

(' c.. t ' \

)

0.rt;e,1e.

/2. '0 ~ X=: k + J(o- ; e eo.",'i -t- 2fl1wJ"""') f~L=: ~.- t L. ~VVf

j~ == C!o-~)e-dj"", {SINwt (-W- 1;~) 1-<0&,,",1:(~"1 - ~"'U-,

dX.( C) \ L- -ci/ZW1,

Jf.:::= - X() -~) ~ e- SlAv<J.rt--..

(6 ) 1'0IIi:. == r LI I-e-eMj",,{u,!«iait~..-U7'LI11/ c.. \ IX(6+)= Xce. Lc.,wdl: 1-2n1W.I~W.:tt)-} k "t

(-e~~ {a6"'d-/:+~;J~ C,LJ.I4+) noUl <.ASe..,

Itm., )( (041:).:=:x.,+ V Itm At L'f/-tPr!-",(s0.,,(eJ.\t ~o 4t-'> 0 .

-d#~s uJA+'(~~ ~i11)~~(~~{uJ +-~j - v::::: X -\.. I I e - AO0 K I~ Itj~T::I

JX(

J)- k.

-a-tj 1»1 Ai +:c -GJj.-/;:~W]0' (I.)W4.

-bJ.t /:,1::. - -x" f11uJe Qn OJ -rY/ e wA-t

{IWv AX ( Jtit ~ i) dt: 41:. -0 + .J--

VV;

~ (~ ((I/LiIv<;//Vw4t

;) ~-I::.~i) wllt:::::- r

I~. I(It: rl.-t

X = c.,c., t- Cl.eII = -C;~k

1'2,.:' -c-(c'-~k 0""::>r, -:>rl-;'rn

~\.) Cl~1:. ~ 00 ~ X--"? 0 . \Jhe(\ di)t?j x=- (:) ~0 ::. C c:r.-t:.h:: en::t: or- -~ -:: ,.,([dz.]t; i If c -fo )

I L "-', \oJ l 1

I ,fc,-::,cJi == 1.1-("1. In (_c'-k,) -tf,CI'\ X::'Cl.~r.-I:

, ~-e.d, rtecJ. 17 ~~:P.-teIS II !'he. 1.5 p1)s',ti If0 ~ ~"'III~'I""r>\)~.l.J;I(M)

if -C.t/c( '> I (ane:{ -t{e/) cMl" OV\ce )Dtie/u.ul'~0 ~eAe cue VlC rt:>s;1:.'ir-0+oVleJ ~

('0) X{o)::::'Xu'";>O . . . X():::'1.+-C~ , C,:::X~-~. dx Ivnpl,e.J Ilhflies

-rI: (\)) -=--Vu Vo=C/i~C2..G. VI):' f,(Xo-e9+rLCz.

C - Vo-x..rl;.- -r1.-rl . - C z. 1/6,-X~(I \n Z ::: '">

c( = Xo(2..-V~ I Vu-'4J2

I:I,-fl L-h V6.~S €ilA i li6r1.iu.nv ) .

.L...JJ

G

"¥o~!lble.. )'fJ .v v '? If;) r1. J II-C'\; V(;;-XI) G ? - x,,/1.toV0 ~I/.ICe. 0 ~ (/ -:>( z..

',r V;, -< X,J"2 ~ tkY'\. V\>-J(b(L.(. 'Io.-Xorl- Cst:~~r,?,-)I

,', \ V. ~ XJ2-}d~~~"~ (~~)clcn~ ~'\eccrt!\r-e..,~~c~ - r-c-~k\

Xo(2-Xo\. J-tY\ )

!a~/- Xc>, -tee. ~"r1t)ret'\e'(ftwe -tf,'$ ~GLtc1cl.:-inde(\<e ~ c/1)Jk IS(here ~It(~&

1~.3 X ::: c e,(;-t:-t-G r"rt:\!-e

xl())-;() "

c\xJi{~1:: Vr;, ';> 0

-c- \'-l ",\.-,

1

(', ::. .:2"",

v:~e,1e. r:;- L-c.ot ,<:: -'\MI\,

12..= ~

0 -=C(+C::~

vt. ::. '1 Ii -+- c:J..r 2-

iYi\Qli~S CL::. -c,

c.\ ::.. V/r,-r2

(0..') X- \/'1;>

(r,-t (2..1::)- - e, -e

('-(2 '

~:!.i.

'I i5 m~re lleo:>.-b"veJ~... r~ e J

rDU.'1~~~c:.tc.~ (l.:,o.Se.JoY"\ ~)

0 Xk~ e"<+ ..

~){ C2--1W\k. ~ c \ \-<\~C :=. C (\- .1,.J, ~ . ,.)c," c::.'"

r.- C/I I .'\J - /tY) r L

-1.. ;:::... ~ .c:C-.\rL. Z -\:'..;6 r, c."

'c.,) v~ \/~ I~rl.-!:. I'I~ )~ 1'\ - t '-' -6-c, \,

ex.cept ~r sn'dll 1:IVhe..S efit.. ";>76r,'t.

:. X % ~er)..-t:; Vfl~~x.~lhu.n,-X. . -(I GtP'1'r'1v.dw

\1'" / ~ qP?rz;AI;?'I'~

~ ~ ~~- -::.--(I C/

. t

/3.i(jt rj::

X ::: CI 8 +-C 2..e lJif&1 e.,

(~ -C+-fCL-

4 l~

I - - .'fIr\..-L..",

rz..-= - C - (Gz-'1..'V1K-L-VY)

os IVU Ib.,~~) r~ CL">"/ -<tYYl~ )

'i'~ '-Ve-0 _.( ...ci. -<.c( Ir /V c/ ~

2.. , - I,'Y\.. -

(9.) SjVe.IVl ,i ,uP ~-rJ'iJ ~ j ,

0(;. /' - Xu'L - v;,'---I- _°12 ~,

Xo -= C( + c 2.-

'Ie. ==c, G +-c2- r2--

. '1_\0I' ."'v J Vo C XC)\ I 0

r. '---I,-..,.J "'0 - - ). /V .-x,

( -' {' ~ t r2... -r;C ~ 0 ( y~. .2.- /- L- II I ".J--fJ )(I -rl.. U J G OAj e" J\ ~er-.. C / ~ Xu

(1) 0::; -~X'- c <IX)J-t. .-.0 ~ Ae- -t -b J &\ ~ 0» e C<.vdo..t" II)( "--) Gl 11 r-s t -0 ref £,1 ~d.. --b d'Y\..;

Iv- p~d: (Q.J j If -t\~ la-l~~ CVWcl(j~\ So ~t

fsklt. - daro~ir-d' 8Xftl1ev1il~ ~K.tS Cl/c:c9 offJ iterv(j -L --si:X ~ c( G(IL:. -:::-CI e, c..-

~~+e£ld ,~ c,x-!, .'. \st-()a~et ~o:bJVV

creJJ.s X 7- (Xo - ~ \e--cJ= ~ -\hIS ~t(~'e<J

n:=:.ifI'l'U IVI ,t IlJ> ~c-L D <JV\J ) c;I tvCG :<ID) -= Xu- 7:JX \ K ( Vc., )") d-i:(OI ~ --S \}o- -(2-

~wev&<-/) 'I! r2. LS"\w~~( ) 1fe/0 XLD)~Xu I( I

~r e,.tt1w~{e- ~ .. "

3-5- 1\ \ Vu( < Xo ~ ,~er0

~ i*- L4-jx,,1 .~ C}>7-'!",k., iAld TfeY\CI ~ Xu I

L~ ) J Lx: - C.\X

V\r) J t.7--:::: - C Ji:2/K:J <)rJ <9-'C

c. I.-- - -;;-,-r::=.x-= t:: t- Ie

1 ~ fwt-h~d~esV\'t ~()- l slovu b-0

0. slo uJ ~~<j)

~s LS CL ~c}ncA,b(e CLf1.f/~XI~Q;--ltdYv &\~

urfevv s [."v:> ~(t"JG fe,LVYlS !?@OY1'i:.

~'irJ V11udv. J:-" vvv~ S~(f.s

em~ .

()'

cI\r----. "" rl- '>..5L r.or

(0L!..CV'.

Q...(..-Jt

I>- rII(:..

rl- 3 l/' ::. 3- -'<-- -..". >-c.§" 0

C'\ ISL f' I 19- *rn0- t='=t...s:t-CJ.!:; :--"r\-,..t -ir- C\ L{;'i I' r j

g + V. 4- + X- '\' f (\ 0(/I

J&("\ Jr- .... Is:z

C> rt.... ;Nr+- -I

3'0 + {. T (;

i 2 l' s::.(..., " 5;.<4..C ><:. /<;;.....-

B ;t Ii II{'.J\'I

P t-t; ..r..- 0 rt- rr (i:{JJ -- \.......-

'-...../..

:x.-

E;L E

""f' ,r

. - . -- O' (b (/'. I

-p cD

- : f- - =- II"5 . cr.;. ,!.L . E ., CD I

jrt II' I' vco -" "C' # - ;)

.., o I CDrt '). CD J 0 VI P-15L-

g A- t: cD" II. f1- I

l./', i"' QJ:ItJ 0 -v + IE l - '<--tJ Q C\ qD -----1

ai. -c::>r- c.. (' 0> Ie- <: -' .Q". \- (? P ::>. l>J<D-' .- V. VI 9.J E

>= D V- EJ=i ft -tL-tr-- \1

o c- V I "-/'"

V (I ilC+