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Unformatted text preview: University of Waterloo
Department of Applied Mathematics MATH 218 — Diﬁerential Equations ‘
Midtérm Examination. October 29, 2004 I‘nStrﬁcmr’: West {formerly M. C, Chidichimo] Tinm: 1 7E, hmlm
Cal'ciuiiamr Aiiﬁwaﬂ. NC} OTHER. AIDS
name (print): . u, . Initials: LD‘ Number: » , Sigﬁature: ‘ Itistr’uéﬁons:
1. Fill if: Hymirnéﬁne; ID iiﬂi’ﬂbéﬁ am} Hm MARKS
’2. 301w: the pmbkmm in the space wovidmi Cmstinae (if; this back 6? the preceding page gnaw}: . . 3. It 1S ésgéntiallthat you eXpI'ain F0111“
metimds‘; and. j‘iiﬁtiﬁ' 3mm C(HHIEIP
signs. I 4,. ﬁfhemver pQSSEbIe? giveihe soiutébn
ta 3. DE expliciﬂy. ' ‘ .C‘ﬁ ; Check ihaf the Examiﬂmim has ‘8.
pages » ‘ ' ' MATH 218 » Midterm Examination Fail 2004 {MARKS} [f3] 3'. (Innsidm “112‘! autmummxs DE
du ,, . .. .r}
_;=. [1.“ 3H,; 1.“
d: I y, H. H: 51} Find the: ('aqni'librhm: salutiom of the DE.
{h} Draw u. phase linen, {:‘aém‘sifymg mu]: (’fi‘itzicai point as mg‘mpmticaliy stable. unstev
bk: or Hﬁilgrﬁalﬁﬁ‘ {(1} Sketch typical soluticm muwes; in the regions in the: ng—pfane detertninmi by the
graphs of the equilibrim‘n ambitions. i
g
i h‘I‘A‘TH 218 ‘ Midterm Examination Fall 2084 b} Shaw Him the foihwwing DE“ ._ V
. V—V 1', J! is homogen‘ems {in t.pr of ﬁrst. order (HE’S) and, using an appmpziattz
substitution. ﬁnd a snl'utiran of the satisfying tin: initiai (:CJI'tditiml 31(1) 2 TE.
Determine the intervai in which the mimic“ is valid. ’h'iAT‘HQAIS‘  Midterm Examination Fall 2004 [7’] :1. {a} A tank contains 1130 ﬁtness 01' pure water. A salt soluhan containing kgg’litre is added at. 3 Haw/“min! and tin2 wellstirred mixture. leaves thr: mka at the sauna
rate. Shaw that shit munum mm in the. tank at; time 1 satisﬁes the DE dm 3 f .
:2 —[_'TR w ‘25) "it." we and solve fh‘riﬂﬁﬁ), ﬂing an appmpriam initial condition. (1)) At. what time ’7' win, there be kg, ni' 54.21111 in the tank? Clive yc‘aur anwer m 3
Sigmﬁcant chglts, {.55 a (hack, Emu. 4 m”; x Arm 4 mm: 3011 vumld expat. 3.0m
answer to be close to 1 mm.) (k/‘hA :(\f\ ~(6Wb H
,A
m V a/jzﬂ
I\U‘f> ,_—1 \s
Cf‘)
l
m
3 MA’I‘H “218 ~ Minimum Exalrnivnracignﬂ 7 FR” 2004 5. Aﬁsnmt’r 1.11m. 21 Sp’ituriivfﬂ raii‘fu‘h’iip {Evé‘zpﬁt’atrzs 21% a rat? pmgmrtimm‘l in 31,25 51162199. area.
H its radius; originally 3mm, and 10 mmnds later has been rammed in 2mm, find an
expression for the. radius of the raindmp at any time. ...
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 Spring '08
 ZHOU
 Math, Calculus

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