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mid1sols

# mid1sols - First Midterm Exam Mal—2403 K4,K5 Instructor...

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Unformatted text preview: First Midterm Exam Mal—2403 K4,K5. Sep 20, 2007. Instructor: Dr. Luz V. Vela—Arevalo.. E amﬂ. gi !‘ aﬁjail. Show all work to get credit! 1. Solve the differential equation dy _ 2ye2 3+2zcoosy d3: _ \$2 Silly—- 62‘“ :[m d/vﬂelenkdl £qu lSpoints (ageihh 0:0de {— (6 ~7< simﬁotj: 0 Th e \for‘ is :xme/i' «u €qu “(23 “+wasj):9,e”m17<9¢ny:§—:X (6 "7‘ W3) ’TW,MF: exists F “a“ ’Wpﬁ 3F: wa5 (land 976. 2:9th ”£3 63 Twin if: {(23621 +9,ng (312‘: 381 He'XW‘jJer‘é) :62 Xd-xlﬂma _,’---I 3‘3 M 36” Macaw = 0 :5 Jr‘s-gonad Samoa. 7) 9‘? a 31x-“ wUl—L‘Q) -: 4: IX my 6:") ¥C33i59>ix®3Q 2. Find the general solution of the differential equation y’ + (x + 2)y2 = 0. Find the solutions correSponding to the initial conditions y(0) = 0, y(0) = 1/4- Where are these solutions deﬁned? 10points Far jewel, inlxtsoid‘h'on r5 jcnrgofiwthx. m“); FUY 'jfo '- l 013:,5 -— ni—zﬂﬁalxu—Qﬁb : ("3’1 (MZMX <2) 3 2 3 gave \ 3'3“ in J— . :4 ° #— .. W“ 5“” 4: r <2 r5 C w 30*m '3 Te CkCCkIF HM aka/um na‘i‘DY “6‘0 (152,55 XL“ wzﬁJﬂ (.1ng rec“) ' 20/1.) 3. Find y(3:) deﬁned by the differential equation d?! 2 _ _ \$84 y/rc_ mdx— y + T112 Ejnthf‘ iii—r: 4, 625/, Whom wi- a 3: MY “We/9‘- dx (9 1n Kg “HOWOUS. \lﬁl S‘J S 4. Consider the differential equation '«vmr'ﬂt'trzqﬂ-Q’ZIU’ é") WITHQ >{ ’2‘!)- ';§m'x+C dy'_ ygul (3) Obtain all solutions. (b) Find the solution satisfying the initial condition y(0) = 75. - d13— y 10 points . (0) Determine the points (a? b) in the plane for which the initial value problem y’ : ( y2 — l) /y, y(n) = b has (i) no solution, (ii) a unique solution, (iii) inﬁnitely many solutions. 15 points 1 "A - (5» ﬁts rt) (233M ’ ”(+6— égg ,L (in). : <7<+ec (3 3m Alta) 5900;, 99394 19) l; ways: salt???“ on manual “an a n mam):¥a§fﬂ"" ‘5 0w 9 Sol ohﬁ‘em. (:73 Q5 W onus 96! (3‘2 300 um lccllx 3“]: + (wﬁﬂl leﬁndedi (:‘W (3\>1/ mxi 5. For the differential equation, ﬁnd all the equilibrium points and uSe linear stability analysis to de« termine their stability. Draw the phase diagram and give a qualitative pictirre of the solutions. %—y3+3y2—10y Let £%\:jb+3ﬂs,ioj :k3(ij+8)(j—1\ 10p0ints %Q—J\lii3v\u~rv\ poin‘is: 33:6) jig Hg) 3": ._ gnaw status!) Mains-is: ?(Csj\:33ll—'ij'(0 7) Mon—gin 7; who 15' stain; P‘tvs): 364040 >0 4 syn—J5 '0 tqg): |Qr—-|"l’(070 f5 31:113 Malibu mm5‘7‘i {OK/e 6. Assume a ball of mass m is thrown vertically upwards from a height h=2m, and it is subject to viscous air resistance (proportional to the velocity v). (a) Write a model that describes the position and velocity of the ball. (b) Considering that the acceleration of gravity is 9.8m]s2, and that the proportionality constant for air resistance It satisﬁes k/m I 5, calculate the ascending time of the ball, just before it starts falling. The initial velocity is v0=10mfs. (c) What is the maximum height? a) l3? : ‘U' 3 Lo) 7' 2 15 points my: : mug v mr (Uto): o <72) (7)" +5rU' 5’0]-3 e‘lt‘N'r/W' (also Worabté) t - 5i“ ‘ st 5e thqreﬁ):~’7?8€; <77 are :iﬂé t—Q ch): e 9 5 4—2 ant); #3355— we“ Sm WCOPJO CW7; @8th «5 , JUNO: -"3—’5§—+[%Ho)g a“ {O Amdwj Wm with) 7'0 (:73 — (,-_ kg: ,- (.7) {:j—‘JS—gvﬂ(ﬁl?/S+IO> , _ it c)’m My” 3M):§UH)CH' :«igg—t ~%(%Eizo)es+c __ V___..(qsg f/OJFC @306); Ht %(7-5 1919165 +2.;— +7§(5-§9+zo) Maxine/taut: 3“”er gwgﬁz) ”éW53‘u‘gs al+l§li§+m> we)? 2 ea 7. For the initial value problem y’ x —:c + 2y, y(0) = l, calculate the ﬁrst two iterations of the Euler method with h z 0.1. ' 10 points 8. A swimming pool full of water also contains pollutants that amount to 10% of its volume. Pure water is pumped into the pool at a constant rate of 7" liters per second, and the mixture is pumped out at a rate of 1' + 2. Write a model that describes the amount of pollutants in the pool at time t and ﬁnd the solution for your model. When will all the pollutants in the pool be removed? VBE lﬂl’h‘al \IuLW as; l‘ialu'xolv 15p0ints 7) we; van—2t mam In we «“1. Let 7m) ti fl/u Mimi cg (scan—Mas, X60): QLVO F "X ‘__ (‘K limo (pm will“) C°”\T“’vc«~at in: T‘ Vanni 'K f, 3-2:: I (left, “' Cora : ’Cr+23 “Gigi: WNW %-:'(H)Ur ; 7((0350NO gﬂtmm SM FSPUQ) At 6?.) Fox :iri—(mzﬂnQ/orat} ell/ac '5? ’ \la'lt *1;— Cﬁ Mic): owraﬂ ”it; A,“ 1 Z 'OlV 0\\JD:C\J:l @cso,t\lo ’ ° K0 :O-'\\lo ‘12 WW mmmmwmmwﬁmm ...
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