Midterm1_key - MATHEMATICS 201 SECTION A01 MIDTERM 1 June 8...

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Unformatted text preview: MATHEMATICS 201, SECTION A01 MIDTERM 1 June 8, 2011 Name: Student Numb er: 0 Duration: 50 min. Maximum score: 30. o The test has 6 problems and 6 pages. 0 Full answers, please. Justify your statements. 0 The Sharp EL—510R scientific calculator is the only calculator allowed. No other aids such as books, notes or formula sheets are permitted. 1. 1. State the order of the difierential equation and give reason(s) for Whether it is linear or nonlinear. (a) y’” + 3:421” + 2y’ = 3x + 1 flin/ Jig/fr" nan - find (60 Mr, WM 3'3 “6 ) r“ , VI l Xxxgl A, V'VLw {J (b) y” + H22 y’ — (sin fly = em 1 3L6 Mfi/ rJY‘Cl/u / C/. 11,. 5'3“.C§L ix, Midterm 1, May 2011 MATH 201 Page 2 2. For the differential equation 5-: = y2(y + 1)(y — 2) (a) Find the critical points, if any7 and draw the phase portrait of this d.e. W510 vb?" wk: 2— (b) Classify the critical points as asymptotically stable, unstable, or semi—stifle. t. > 0. A {kin} FOL (fife {’ (AS 46 At in m , SOL V,‘ 31m LI, F1111 a} ' “as; '1 ‘3 Us 3‘ ’f) ‘3 a :7 N5 3,- I: ) (JAVA (c) Roughly draw the solution curve, Without solving the d.e., for y(0) = 1/2. Midterm 1,. May 2011 MATH 201 Page 3 3. (a) Show that the differential equation (36% + 11:) dcc + 8”” dy = 0 is not exact. fl _ 3,5,; A b. i 6,5; so 0v» 3‘0. LC: fl-Jf 12,4 qucf OWL (b) Solve this d.e. by determining an integrating factor which makes it exact. Hint: You need to check if the quotient (My — Nx)/N depends only on 3:, or if the quotient (Nx ~ My)/M depends only on y.) iwd‘ #3 w ,‘ c1" J1 C‘j“ a Z“ ,3 ler/ {It-Z) {'0le A?“ 0/ a? Q. 0 at w h”. JG EVA ,6 9‘9! A 3K " iii m i :3 WLW fl SJVJQ“ 1“ J f e, “X 3 ’” (3 23‘ * 7L 9 ) $ 7) . a“ { k2 2A ) C/W U» 7 i M 0/“ I Db i w) - B L» :5 Q; m 1 5K i c,“ )?G 31 (J "fi 03 '3 _‘, E i : b/i ¢> er 2: e as 0‘. l C so 2 k4\)’ l K g) O 5* "A it} 5 S C { /‘€r 4 L Midterm 1, May 2011 V MATH 201 Page 4 [6] 4. (a) A population of trout in a lake is assumed to grow exponentially. Suppose that at this moment there are 1000 trout in the lake. Also, suppose that next year 1200 trout will be in the lake. Find P(t) using the appropriate model. ‘ [if _ kt JP 3 1C ,9 a: Pill) T3 [00 e’ s '00:} {7/ It k—iZ. Lajmhflz) (b) Suppose that 100 trout will be harvested from the lake at the end of each year. That is, suppose that one year from now, 1100 trout Will remain after harvesting. Find P(t) using these two facts and the appropriate model. Hint: Assume that k, the growth constant, is the same as in part (a). JP who!) : mm) 9"” o"t # 2/9 Man” 3 Jaw 1W at 3) 7+ ’ i/ m. 1 WM € ~_ e .. 1- , ,9 [n(| L)f ' F e/lfl( Z) t. finLLZJ “(live (00 ., *9.) (2’ P: “H30 12, plnfll) lo (11);!» {.3 P, 100. + C a ln(l~1) 109 Phh HM a“? C : Moo — 177M) Midterm 1, May 2011 MATH 201 Page 5 [5] 5. (a) Show that the d.e. ' A X y(1ny —1na: — 1)d:z: +03dy = 0 is a homogeneous, and states its degree of homogeneity. {X I (AM— "szu/ to 5h5w' “MN/8‘ firik);(n6)x f. U/bl(l"lbb) i Nhhm) ’( ) MSV8L\A&)'\) .13 fl(‘11,40)c ta (1% 61:9“ 5 x N s w Nchflkby ,, h ilk NINE) 56 W‘ Viifl‘b‘d“, 0”" (b) Solve this d.e. by knowing that it is a first order homogeneous equation for an initial condition y(6) = 1. ,Ejjl/ r Jr; at"? \“ \*\§ BLALnu I “j c_ C «12...; 2 MW ' MD {—3 ‘A \ .- C 75—) : C" e” \n V\ l" ) A“? C - ’i Midterm 1, May 2011 MATH 201 Page 6 [6] 6. Solve the IVP d £+xy=xy3fi y(0) = _2 and write the solution explicitly. . \i > ’ .fl) [3 OK [BI/(flair); 9/,e‘ wwax n .. ’ 3 kn ’1 dz: ‘ ’2" 09m 9 “:1 —\ 3 04 / u ’_ > \b c N _ / \a 3 7' JL -L 1/; i ._.———7\ a 7\ 2:) " Ju-Jr Wm 3 W 2 (/K \67’ L a (/V‘ —- 11 \A "11‘. /L"‘9 f L . SPL‘,“ J,“ i 3 1 RJ“ I ')L SE ,e/L“L I‘Akc ‘QKLKC—{fl‘f {i J: a .2 «2/ L Z 2 ,l— / J "*2 tr, ’1’ pt -: B ~'L% e, (V‘ ’1 [a (A 3 - 7.x 2 79 Q] C” 12 I ’ C 2.. “ 6 “f l 2. '- 1 i): z .——> \A 3 \ C e/ a, L S \—i £3 0 7, VA :7Li't'ics‘3CSé Ts?— ‘1 ‘1 1") LL: 6r I) “5 v ~1412. . n . , n KO) 3 _ 1 N we, m Mud comm u m ...
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