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Unformatted text preview: Math 550:391 ‘ I +,
Sample Exam Name: 0 U» lows IIIClass Part Print your name neatly. If you forget to write your name, or write so sloppy that I
can’t read it, you can lose all possible points on this exam! Answer all the questions
that you can. Circle your answer. WARNING: This is a closed—book, closed—note,
no—caicutator exam. Exam format: The exam has been divided into two equally weighted parts: Part 1 is
elementary concepts problems. Part 2 is intermediate level calculationbased problems. Part 1 consists of 16 elementary level problems that test your understanding of the basic
concepts. Each problem is worth 1 point. There wiil be no partial credit on any of these
elementary problems. NO EXCEPTIONS! Do any 15 of the 16 problems. You may do all
16, but i will stop grading after you get the first 15 correct. in other words, you cannot
earn more than 100% on this exam. Part 2 consists of two calculationwbased questions with multiple parts. The value of
each part is displayed next to the probiem number. ' You must Show your work. You will not receive credit for lucky guesses. Show your
work as clearly as you can: if i can’t understand how you got an answer, I will not give
you credit for it. Grading Scheme for 3 point problems Each problem wiil be graded according to the foliowing scheme: Minor algebra/ calculus mistake with a correct approach: 2 out of 3 points given Major algebra/caicuius mistake with a correct approach: 1 out of 3 points given
Wrong approach: 0 points given The diiierences between major and minor mistakes wiil be determined by the instructor,
not the student! This algorithm will be strictiy enforced. No exceptionsl DO NOT WRITE ON THIS PAGE IN THE SPACE BELOW 1...__ 2......”— 3_.__ 4W 5—«
6~__.. 7..._.~ 8_____H. 9m...— 10_.___
11....— 12W 13...._ 14% 15,—.
16 _._._ 17a m...— 17’b __..._ lSa m 18b m_._ 18c _...__.. Solid cl 0+5 0 :: 5415,15 ﬁxed Phj
Math 550:391, fall 08 Sample Exam Wayne Hacker 2008. 2 { Le+ Aer/ow one o —= “whiskH'er pm In problems 15 you are given the graph of :i: m f on the mowplane, Where v m :f: is the
velocity (rate of change of Your job is to draw the flow field superimposed along the
xaxis. Be sure to label all fixed points as stable, unstable, or halfustable with a solid dot,
hollow dot, or half—hollow/half—solid dot. Use three arrows close together >>> to denote
the ﬂow direction in each region bounded by fixed points, and / or a ﬁxed point and inﬁnity. Problem 1: Figure 1: The Slope ﬁeld for the ODE is f(:r) : ~:r(1  ' Problem 2: 1.0 “1.5 —1.0 Figure 2: The slope ﬁeld for the ODE is Hm) = —(:13 + 1):r(:c —— l). Math 550:391, fall 08 Sample Exam Wayne Hacker 2008. 3 Problem 3: Figure 3: The Siope ﬁeld for the ODE is f = 562(93 + 1). Problem 4: TM‘J'ArmN +56. ‘F’ow “GEM I'gr Cm?— ferlrork cmA repeai‘ “1.0 ~15 Figure 4: The Siope ﬁeld for the ODE is f = COS 3:. Math 550.391, fall 08 Sample Exam Wayne Hacker 2008. 4 f’roblem 5: In this graph be sure and label where the ﬂow slows down. V c. z ValetHy
1.2
1.0 / “(ﬂew “Winj
0.8 0.6 0.4 slower mow'n
0.2 J 5 0.5 1.0 1.5' 2.0 mecr _0_2 Flow {$63M .I'I ainyd‘ mow'n “fb’fhenf law!» Her/<3ng w ~H.e
Mtrouk H’ 3a+s.:fn (ff; I+Legamés GXPMen‘hkﬂy £3093?
Figure 5: The slope ﬁeld for the ODE is f = 8“”.
i he HM" b““s émifcahe unﬁmhle fawn“ cm 44'“: hf” :‘ﬁ'w‘a ﬁxed FG‘H“ Solid be”: indi'cgda s’f’abte fad—Hon: on “H161 MH£>$+4ME “G‘XE’d path\ In problems. 68 you are given the graph of the potential function V(x) corresponding to an ODE of the form: z'c = f (:13), Where f m ~g. Use this graph to draw the ﬂow ﬁeld.
Problem 6:
1.0
'The. «aircs 6F +58 fowg‘q‘ ﬁn —0.5 —1.0 ' 1 1
Figure 6: The potential for the ﬂow ﬁeid for the ODE is V(m) m —&x4 + 5:132. Math 550:391, fall 08 Sample Exam Wayne Hacker 2008. 5 Problem 7: ‘yE' ;“°“(\ Figure 7: The potential for the ﬂow ﬁeld for the ODE is V(:c) = —:c —— cc. Problem 8: ~10 1
Figure 8: The potenﬁiai for the ﬂow ﬁeid for the ODE is V(3:) m 13:4  $272 — Math 550:391, fall 08 Sample Exam Wayne Hacker 2008. 6 In problems $41 ﬁnd all of the ﬁxed points to the ﬁrstworder autonomous ODE and do
a local analysis to determine the stability of the ﬁxed points. Recall: Let 32* ﬁxed point to the ODE 9': m f Then, provided that f’(a:*) ;£ 0, if f’ > O, 33* is an unstable ﬁxed point
if f’(a:*) < 0, $* is a stable ﬁxed point *2...“
13roblem 9: $3241.93), 5.3+ ﬁx) a: x(:——x) z xwx‘“ (1w mm x .... of)
Fixed Points: 0” 'ac/{OJ :1 > o :27 unrl'dtle
Stable Points: l Unstable Points: Halfmstable Points: Nah (no double reﬁt?) 523' 5300 :3le 55—0 :22) )(."‘~‘lfﬁ’l‘J nnéZ,n d2” (Mm0M “
f’(n1f)‘:: can: [m0 2: Pr0blemll,{}:w:i15"sin :3.
Fixed Points: ﬁn»an nel—
Steble Points: mi’J n (Mel ‘ Unstable Points: HT?) V1 €03" Half—stable Points: norm SeffCKl": "'3 a w»
Problem 11: :2: =1— €32. a, . o :1..([vx +
laylal’ﬂcfométﬂbo‘d J 5300 .— X1+8Cxqj Fixed Points: 0'
Stable Points: None
Unstable Points: Mane, ‘ Hallistable Points: 0 Math 550:391, fall 08 Sample Exam Wayne Hacker 2008. 7 Problem 12a: :‘c m 0053: 2:53“)
V00 2 — Scans! AK : “jelshnx = «smx +V° (can 5+0? hare ‘1 3 59+ WE’eﬁf'ﬁ‘ﬁcﬁ Pghen‘Ix‘o,‘ 5: O 7': > VD $0 (+1455 9+3? 1'! 100+ haterrary) ’ :V(x):—5'nx Problem 121): $ = — sinh :c::§(K)(Note: / sinh :1: dcc w coshzz: + constant) V(x):: *jfbﬁMx .7. ~S~s£nkx<ix a: mam : coshx +vo. Math 550:391, fall 08 Sample Exam Wayne Hacker 2008. 8 Problems 1344: (Impossibility of oscillations) Recall: In class we showed that the
solutions to a firstorder autonomous equation of the form :is = f are always mono—
tonic on the real line. That is, if we take any initial condition between two consecutive
fixed points, then the solution will always move to the left or to the right. AnsWer the
following true / false questions. Problem 13: The solutions to the systemgzt w cosa: cannot oscillate in time.
. Thai: 4'5” 6. "ﬁ'rffvonlrJ" mutton omoux xys+emwt+h .Srwo‘Ha ‘53“,
False (encie one) ‘ Problem 14: The general solution to the harmonic oscillator i3 + (.0233 = O‘is given by
cc(t) = (:1 cos(wt) + Cg sin(wt). This Solution cannot oscillate in time because the system is autonomous. :[3— Aces use; “1+3 Bu'l‘ ‘HM'K [3“ at 73‘“ order qtil‘bmumou ODE
True (Circie one) QWX our FreeF {5” only Valitbl‘lhrﬁ'gm (ﬁler “Whmmws ODE. Problems 1516 (Existence and uniqueness) Consider the ﬁrst»order autonomous IVP:
m' = f(m) with 17(0) m 330. Theorem 1 (Existence and uniqueness theorem for a ﬁrstorder autonomous ODE Let I g R be an open interval on the m—axis. If f’ is continuous on I (a subset of the
Lit—axis), then there is an interval on the taxis (t9 —— T, tg+ﬂ centered at to for some 7’ > 0
and a unique function a: = 33(t) deﬁned on (to — T,t0 + ’r) (the domain of the solution and contained in the range of 1c, denoted Rm g I satisfying the ODE. Using the theorem, determine if the foilowing ODES have unique solutions. i’robiems 15: 51': m 36%; 33(0) = 0. Tl“: 'H‘ec’f'ﬁm "all: +0 halal he“? becau‘e 3:60 :: x ’3 :3”; fix) : % xng’ﬁ has “Sm‘hgwlariﬂ oil" x z: 0 J w Mali Es ad— +53 . I t
Paint" 0? ow!" I_C , 3:“) Sol“ neecl he‘i‘ be unique. Anti J we doﬁ‘ilnw uumquesolr,‘
((Non uniquesomrl No’h'ca ‘l‘ho‘l' X150 (5 :1 Sci“ +o WIVP above. 73%“. 0k 2“ morn:an
Sol“ we {win m’ie— “Hua ODE Lulu S‘Cfdrn'lﬂcono't: Variables.
item A” M x“ Vaax ran Mesa ext? == be #1231; % 017520” to :> f" 30‘ 3/
m (‘Zﬂd Sol") $>X££Lzz Problems 16: True or False: Since f = 3:2 is a smooth function on the real iine, it i 0 follows from the theorem that the solution of the IV? 9': m 9:2 with $(G) = ace exists and
is continuous for all time t and for every initial condition. You must justify your answer to receive any credit.
Hint: It’s not hard to solve this equation using separation of variables. In fact, you’ve done in your homework. I ' I t I __ n I “A
7711‘: Sol“ has gyrosfun ecu: swam (LPU'QnHe—‘hme He)qu )jm) ﬂfzﬁaeﬁfxg m ) '77“? clear no‘l' whining ‘Hie “fheurem Jbeﬁausehi‘l' only guarantee; q So‘hon Some :nmd 0‘“,
time. 32+ Mates to frame: eta “Hie Sol .5 mm vs >0. 1L :3 k ~ — JaeM 6 Check. gézxz' w’x”Z‘AK:Ah«i~—>x‘=£ to xfoirzxo Xe ° 0 Xe , "1 ....._ l r ' I x0e): ribnan (emsmm new “PL Math 550:391, fall 08 Sample Exam Wayne Hacker 2008. 9 Probiems 17 (6 points total) (a) (3 points) Consider the system :i: m "m: + :53, Where 7" > 0. Show that 03(t) we 00 in
ﬁnite time, starting from any initial condition 339 > 0. Hint: Notice that :3: > O and m > 0 (Vt 2 0). You shouid be able to find a simple
” comparison” system for which this is true. (b) (3 points) Suppose a stabiiizing nonlinearity is added. In particular, consiéer the
system
:2: = rm ~i 9:3 w m5 . Is it still true that 513(t) —> $00? To receive any credit, you must justify your answer with a calculation. A yes or no answer Wilt receive a zero. No. X Lt} => {‘53 L251 a: {Taco Warning: Do not try to solve the equation exactly! ‘9’" My X0 >°' Hint: What’s the large a: behavior? Try using an approximation. Also, you may ﬁnd the graph below to be helpfui. )2 :3 ﬁx; ﬁr [We x a“) I"? X" >> Ifi‘hen XL—U Jams“
mmoﬁm‘mllyqn Kira, LgL VvoxL‘f
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2.0 Figure 9: f(:c)m:v+cc3——:I:5, forr=1. 3 3 Shae 2(0):)! F?OJCU,LAJ‘3 K>0JWC haJ/‘e— c o 3 \ Re I ﬂy mu” Coumh“ 4+!eommj [41 but cmwhow has «can *‘thﬂb oqujJrhen Jo aloe: ).(::\f‘“>< +X3 ' 5 wa—
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£9 70“ =_ e» as: time aw cf ° Math 550:391, fall 08 Sample Exam Wayne Hacker 2008. 11 Problems 18:(9 points) Consider the equation a": =r33—~ atln(1 +33)
with parameter r, on the interval (—00, 00), Where f (LII, r) = m: — a: 1n(1 + 93):: x (f—— ln(l+X)) F
:e xcrwx) (22W )
(lam We a. «amass imam ) (a) (3 points) _
Step I: For each real value of r, ﬁnd all the ﬁxed points of this system. You should be able to find explicit expressions for these.
Step 2: Find the critical values (mare) where the bifurcation occurs. Hint: set 22(1) (’1‘) = 33(2) (b) (3 points)
Step 3: After finding the point (we, re), do a local analysis of the ODE to determine the
normal form for the bifurcation. lise this analysis to determine the stability of all of the ﬁxed points for both cases: r < re and r > re. Recall: The normal forms are: X :: R :I: X 2 (saddlepoint bifurcation)
 X = RX (1  X) (transcritical bifurcation) (c) (3 points) Step 4: Now put together all of the information that you have: :rmﬁ'), for 2' = 1,..11, (n roots),
and the stability information about each branch, to sketch the bifurcation diagram (is,
the plot of the ﬁxed points as a function of r with the stability indicated by a dashed line for an unstable branch and a solid line for a stabie branch). same em31 with Seesaw) $0.
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