Test1Solutions - Math 306— ”Test 1 Printed Name[Alf E...

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Unformatted text preview: Math 306— ”Test 1 .. . .. .. Printed Name: [Alf E Please carefully work all of the following problem(s) You must SHOW YOUR WORK to receive ANY credit! The Printed Name question is worth 5 points. 0 H indicates a “By Hand” problem where technology is not allowed; o '11“ indicates a “Technology” problem where youm ax use Mathematica or a calculator. You must still show the steps taken. For example, f: (1 —:1:)7 dz— m —8, 192 is valid. Giving the answer -—8, 192 without the integral is not. a Do not use DSolve or NDSolve unless otherwise directed (except for checking your work). Find general solutions for differential equations and explicit solutions for initial value problems. 7 71? 2 0 i ‘3 i225!“ Ellinlxi 1—1 23y+y3103fu7>3 + {313 :S/n (r e 11. £391]: %;.F=S W3: g'fl'icl 31M: Sfiic'fl’; Dr. Emmert Spring 2010 I of IV Points Earned i:::| of 21 Math 306 — Test 1 I Verify that the difierential equation cos(x) )+ mg) + (Z + By) 5-: =30 is exact and then solve it. Wml = W (x) will Mal“) ‘53 ”J33 35.)) no em T. M L : g- ‘5 N l hl" 1/ (7333 (j +8) Xx} _ ‘3 :{33:3’€‘50)3: (33 +5,- ‘lWl: will ' rymyfij) ; Jmlk} +9‘l”‘(17l‘lej:c Hhcg3: Smwfilcly SEMbcthlyflolx 7 Imfx) «3 «Hal +3 (33‘ - "K “alkfihrl‘obfivfl : (lg. lflfl‘fl)‘ 9‘ 3.6 LE) qu] : Use substitution to transform the following differential equations into something a. little more friendly. Do not solve them, just write the transfonned difirerential equatian! r .. 33‘ HATPts Iy'wy+2\/§c_fl. :3) :3 ~ i— "l" 2 1-; (Homfibmj)‘ "b“ all!) :0! l—‘l 4" ) vwfl.“ dml‘lu dfii‘ )c . d ,_ 'y'fifi-l'v - val?2 ELL “23—31 - 3 a mfuyrflwwaezm) M J %;:e ’20.!" — 2(v+9< 33”) airman“. V V” 3*.“ ‘- d5 :4 chL-—3 ‘qolg/ file .3. (73:3 ,333’3 , a; f; (33% V m: Points Earned :l of 28 II of IV Dr. Emmett Spring 2010 Math 306 — Test 1 Verify that y(:c) = Acos(3r) + Bein(3m) is a two parameter family of solutions to the differential equation 3;" + 92; = D. j, 23A Im( 37:) 1‘38ij (3X) ‘3" " “3A [mm] s cm mm) / gm : [-iaraoxmmmtm HHMM + Balm] = O Sketch likely solution curves through the additional points marked in each slope field. Direction Field E: A deer population P(t) in a small forest satisfies the logistic equation fl—f m 0.0225P— 0.0003132, P(O) = 25. Construct a slope field and appropriate solution curve to determine the amount of time, in months, for the number of deer to double. Attach the picture to your test. I: 60 “trill Construct a slope field (or use the same one in the previous example, if appropriate) and appropriate solution curve to approximate the carying capacity of the deer population. Attach the picture to your test. Dr. Emmert Spring 2010 III of IV Points Earned [:1 of 28 Math 306 — Test 1 Consider the initial value problem if; = 9 — 42:2, 3(0) = 0. Determine the equilibrium solutions (based upon the critical points). 5 Determine the stability of the equilibrium solutions using a phase line diagrams r {3) i J (-) C +3 C") . .1 W .. 3/ 3/: 7, rs 'fl\ meme IL-Jslg Determine the stability (stable, semistable, unstable) of the equilibrium solutions to the differential equation -—— = 32(32 d 9). Include any pictures that you use to test stability. 5li I): 04L”) Construct (do not solve!) an initial value problem to model the rate of change of velocity for this situation: “Suppose a crossbow bolt is shot straight upward with an initial velocity of 49 meters per second at ground level. Assume that air resistance is proportional to velocity and use 9 for gravity." ”5%: Mill 3,WIJ it‘ll fz-YN VG :Ll'fimfkt. 7,‘ __ I?“ Ch} rfl e: la} Points Earned l::I of 28 IV of IV Dr. Emmert Spring 2010 2 I Test! Sofutions. nb Problem 6 The deer problem. hubbn16A:'flfisaflowsustoapmnxhnamthefimeunfi]popumfiOHSOisrmmhed points = {{D, 25}}.: f[P__] :z0.0225*1= — 0.0003 * 9‘2; Show[ VectorPlot[{1, f[P]}, {t, —0.1, 100}, {P, —0.1, 100}, FrameLabe1-+{t, P}, Axes—iTrue, VectorScale—9{Tiny, Tiny, None}, VectorStyle—)Gray, StreamPoints—+{points}, StreamStyle~afBlue, Thick, "Line"}, StreamScale -) Full , (* Generates solid curves for sample solutions *) PlotLabel-+"Problem 6A", Epilog—e {PointSize[Large], Point[points]} (* A good way to indicate the initial conditions *) I, PlotISO, {t, O, 100}, PlotStyle—>{Red, Thick, Dashed}] I Problem 6A F“ “ '1 ’ ' ‘T' ' "'T’ ' ' T"" 'TW'WY—"fi'V—mfirfi'“TMfi'YWMI'mM'M'!'“m‘mr‘m‘T‘ - W?) Imk ‘ ‘a '0 ~kJ ;. 4* 5 1* ~ ~ "! wk * v m _w% r -§ § - a "j (30 » » w— , , . .. T we , - , .,l g 0 v . g |_ 4- a l I i n in ,, iii, ,1, 1, A, J 4, x l ! | 1 l J 0 m 7" 7m 7’ 7 mi m 1% Test] Solutionsmb i 3 Probfem 6B: This allows us to estimate the carying capacity. 0.3,, points = {{0, 25}}; f[P_] :20.0225*P ~ 0.0003 * 9‘2; Show[ VectorPlot[{1, f[P]}, {t, —0.1, 250}, {P, —0.1, 100}, FrameLabele {t, P}, Axes—’True, VectorSoale—3{Tiny, Tiny, None}, VectorStyleaGray, StreamPoints ——) {points}, StreamStylea{Blue, Thick, "Line"}, StreamScale —-> Full, (* Generates solid curves for sample solutions *) PlotLabel —) “Problem 63“ , Epilog —~> {PointSize [Large] , Point [points] } ('k A good way to indicate the initial conditions ir) . Plot[50, {t, O, 100}, Plotstylea {RecL Thick, Dashed}] 1 Problem 613 fiTfi ' "Y—T”WWMT"MW"-‘TMMDT"MT‘ [ I l I u I I 1 I ‘ a 1 80 . ,, “ (:0 7 _ j - r 4U — —; 1 . J' 20 -- U m gildmeflmi. .....J..__J .L -1 - J l l I l s i z \ L.__l .1 .LMW U 50 100 150 200 4 E Test} Saturtons. nb Problem 8: Determine the stability ofthe equilibrium sotutions to the dméremt'at equation f[x__] := x*2*(x*2—9); oritNumbzle. Solve[f[x ==0, x] Show[ VectorPlot[{l, f[x]}, {t, —0.1, 2}, (x, -5, 5}, FrameLabel—tft, x}, Axes—aTrue, VectorSoala->{Tiny, Tiny, None}, VectorStyle—aGray, StreamPoints->Automatio, StreamStyle—+{Black, Thick, "Line"}, StreamScale—&Fu11, (* Generates solid curves for sample solutions *) PlotLabel—9"Problem 8“(* A good way to indicate the initial conditions *) . Plot[critNumb, {t, —0.1, 3}, Plotstyle—+ {{Green, Thick}, {Red, Thick}, {Red, Thick}, {Blue, Thick}}] Problem 8 i ,, ,. % ...
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